:: ORDINAL6 semantic presentation

REAL is set
NAT is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite cardinal limit_cardinal Element of bool REAL
bool REAL is non empty set
omega is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite cardinal limit_cardinal set
bool omega is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
{} is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty finite finite-yielding finite-membered cardinal {} -element Ordinal-yielding increasing V71() decreasing non-decreasing non-increasing Cantor-normal-form set
the Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty finite finite-yielding finite-membered cardinal {} -element Ordinal-yielding increasing V71() decreasing non-decreasing non-increasing Cantor-normal-form set is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty finite finite-yielding finite-membered cardinal {} -element Ordinal-yielding increasing V71() decreasing non-decreasing non-increasing Cantor-normal-form set
1 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal Element of NAT
2 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal Element of NAT
3 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal Element of NAT
0 is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty finite finite-yielding finite-membered cardinal {} -element Ordinal-yielding increasing V71() decreasing non-decreasing non-increasing Cantor-normal-form Element of NAT
a is set
a is set
On a is set
sup a is epsilon-transitive epsilon-connected ordinal () set
a is set
U is set
g is epsilon-transitive epsilon-connected ordinal () set
sup a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
g is set
a is () set
U is Element of a
a is set
On a is () set
U is set
a is () set
sup a is epsilon-transitive epsilon-connected ordinal () set
U is set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
a \ U is Element of bool a
bool a is non empty set
g is set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
a \ U is Element of bool a
bool a is non empty set
a is set
RelIncl a is Relation-like V34() V37() V41() set
U is set
RelIncl U is Relation-like V34() V37() V41() set
g is Relation-like Function-like set
o is set
g . o is set
f is set
g . f is set
field (RelIncl a) is set
field (RelIncl U) is set
dom g is set
rng g is set
[o,f] is set
{o,f} is non empty finite set
{o} is non empty trivial finite 1 -element set
{{o,f},{o}} is non empty finite finite-membered set
[(g . o),(g . f)] is set
{(g . o),(g . f)} is non empty finite set
{(g . o)} is non empty trivial finite 1 -element set
{{(g . o),(g . f)},{(g . o)}} is non empty finite finite-membered set
a is () set
RelIncl a is Relation-like V34() V37() V41() set
U is () set
RelIncl U is Relation-like V34() V37() V41() set
g is Relation-like Function-like set
o is set
g . o is set
f is set
g . f is set
field (RelIncl a) is set
field (RelIncl U) is set
dom g is set
rng g is set
c is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
a is set
U is set
[a,U] is set
{a,U} is non empty finite set
{a} is non empty trivial finite 1 -element set
{{a,U},{a}} is non empty finite finite-membered set
g is set
RelIncl g is Relation-like V34() V37() V41() set
field (RelIncl g) is set
a is Relation-like Function-like T-Sequence-like set
U is Relation-like Function-like T-Sequence-like set
a ^ U is Relation-like Function-like T-Sequence-like set
dom (a ^ U) is epsilon-transitive epsilon-connected ordinal () set
dom a is epsilon-transitive epsilon-connected ordinal () set
dom U is epsilon-transitive epsilon-connected ordinal () set
(dom a) +^ (dom U) is epsilon-transitive epsilon-connected ordinal () set
g is set
a . g is set
(a ^ U) . g is set
a is Relation-like Function-like T-Sequence-like set
U is Relation-like Function-like T-Sequence-like set
a ^ U is Relation-like Function-like T-Sequence-like set
rng (a ^ U) is set
rng a is set
rng U is set
(rng a) \/ (rng U) is set
dom (a ^ U) is epsilon-transitive epsilon-connected ordinal () set
dom a is epsilon-transitive epsilon-connected ordinal () set
dom U is epsilon-transitive epsilon-connected ordinal () set
(dom a) +^ (dom U) is epsilon-transitive epsilon-connected ordinal () set
a is set
b is set
(a ^ U) . b is set
y is epsilon-transitive epsilon-connected ordinal () set
a . y is set
y is epsilon-transitive epsilon-connected ordinal () set
y -^ (dom a) is epsilon-transitive epsilon-connected ordinal () set
(dom a) +^ (y -^ (dom a)) is epsilon-transitive epsilon-connected ordinal () set
U . (y -^ (dom a)) is set
y is epsilon-transitive epsilon-connected ordinal () set
a is set
b is set
a . b is set
(a ^ U) . b is set
b is set
U . b is set
y is epsilon-transitive epsilon-connected ordinal () set
(dom a) +^ y is epsilon-transitive epsilon-connected ordinal () set
(a ^ U) . ((dom a) +^ y) is set
a is epsilon-transitive epsilon-connected ordinal () set
epsilon_ a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
U is epsilon-transitive epsilon-connected ordinal () set
epsilon_ U is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
a is epsilon-transitive epsilon-connected ordinal () set
epsilon_ a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
U is epsilon-transitive epsilon-connected ordinal () set
epsilon_ U is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
a is () set
union a is set
U is epsilon-transitive epsilon-connected ordinal () set
a is Relation-like Function-like Ordinal-yielding set
rng a is set
a is epsilon-transitive epsilon-connected ordinal () set
id a is Relation-like a -defined a -valued Function-like one-to-one V26(a) V30(a,a) Element of bool [:a,a:]
[:a,a:] is Relation-like set
bool [:a,a:] is non empty set
dom (id a) is set
rng (id a) is set
a is epsilon-transitive epsilon-connected ordinal () set
id a is Relation-like a -defined a -valued Function-like one-to-one T-Sequence-like V26(a) V30(a,a) Ordinal-yielding Element of bool [:a,a:]
[:a,a:] is Relation-like set
bool [:a,a:] is non empty set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g is epsilon-transitive epsilon-connected ordinal () set
dom U is epsilon-transitive epsilon-connected ordinal () set
U . g is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
U . o is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
id a is Relation-like a -defined a -valued Function-like one-to-one T-Sequence-like V26(a) V30(a,a) Ordinal-yielding increasing non-decreasing Element of bool [:a,a:]
[:a,a:] is Relation-like set
bool [:a,a:] is non empty set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g is epsilon-transitive epsilon-connected ordinal () set
dom U is epsilon-transitive epsilon-connected ordinal () set
U . g is epsilon-transitive epsilon-connected ordinal () set
U | g is Relation-like rng U -valued Function-like T-Sequence-like Ordinal-yielding set
rng U is () set
o is epsilon-transitive epsilon-connected ordinal () set
id g is Relation-like g -defined g -valued Function-like one-to-one T-Sequence-like V26(g) V30(g,g) Ordinal-yielding increasing non-decreasing Element of bool [:g,g:]
[:g,g:] is Relation-like set
bool [:g,g:] is non empty set
dom (id g) is epsilon-transitive epsilon-connected ordinal () set
dom (U | g) is epsilon-transitive epsilon-connected ordinal () set
dom (U | g) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
succ a is epsilon-transitive epsilon-connected ordinal non empty () set
{a} is non empty trivial finite 1 -element set
a \/ {a} is non empty set
y is epsilon-transitive epsilon-connected ordinal non empty () set
c is epsilon-transitive epsilon-connected ordinal () set
(U | g) . c is epsilon-transitive epsilon-connected ordinal () set
the epsilon-transitive epsilon-connected ordinal non empty () set is epsilon-transitive epsilon-connected ordinal non empty () set
id the epsilon-transitive epsilon-connected ordinal non empty () set is Relation-like the epsilon-transitive epsilon-connected ordinal non empty () set -defined the epsilon-transitive epsilon-connected ordinal non empty () set -valued Function-like one-to-one T-Sequence-like non empty V26( the epsilon-transitive epsilon-connected ordinal non empty () set ) V30( the epsilon-transitive epsilon-connected ordinal non empty () set , the epsilon-transitive epsilon-connected ordinal non empty () set ) Ordinal-yielding increasing continuous non-decreasing Element of bool [: the epsilon-transitive epsilon-connected ordinal non empty () set , the epsilon-transitive epsilon-connected ordinal non empty () set :]
[: the epsilon-transitive epsilon-connected ordinal non empty () set , the epsilon-transitive epsilon-connected ordinal non empty () set :] is Relation-like non empty set
bool [: the epsilon-transitive epsilon-connected ordinal non empty () set , the epsilon-transitive epsilon-connected ordinal non empty () set :] is non empty set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is Relation-like Function-like T-Sequence-like set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
the epsilon-transitive epsilon-connected ordinal non empty () set is epsilon-transitive epsilon-connected ordinal non empty () set
id the epsilon-transitive epsilon-connected ordinal non empty () set is Relation-like the epsilon-transitive epsilon-connected ordinal non empty () set -defined the epsilon-transitive epsilon-connected ordinal non empty () set -valued Function-like one-to-one T-Sequence-like non empty V26( the epsilon-transitive epsilon-connected ordinal non empty () set ) V30( the epsilon-transitive epsilon-connected ordinal non empty () set , the epsilon-transitive epsilon-connected ordinal non empty () set ) Ordinal-yielding increasing continuous non-decreasing () Element of bool [: the epsilon-transitive epsilon-connected ordinal non empty () set , the epsilon-transitive epsilon-connected ordinal non empty () set :]
[: the epsilon-transitive epsilon-connected ordinal non empty () set , the epsilon-transitive epsilon-connected ordinal non empty () set :] is Relation-like non empty set
bool [: the epsilon-transitive epsilon-connected ordinal non empty () set , the epsilon-transitive epsilon-connected ordinal non empty () set :] is non empty set
a is epsilon-transitive epsilon-connected ordinal () set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U | a is Relation-like rng U -valued Function-like T-Sequence-like Ordinal-yielding set
rng U is () set
dom U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
dom (U | a) is epsilon-transitive epsilon-connected ordinal () set
(U | a) . g is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
(U | a) . o is epsilon-transitive epsilon-connected ordinal () set
U . g is epsilon-transitive epsilon-connected ordinal () set
U . o is epsilon-transitive epsilon-connected ordinal () set
a is set
On a is () set
RelIncl (On a) is Relation-like V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
a is () set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl a is Relation-like V34() V37() V41() set
order_type_of (RelIncl a) is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
a is () set
RelIncl a is Relation-like V34() V37() V41() set
U is epsilon-transitive epsilon-connected ordinal () set
a is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty finite finite-yielding finite-membered cardinal {} -element Ordinal-yielding increasing V71() decreasing non-decreasing non-increasing Cantor-normal-form () set
On a is () set
a is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty finite finite-yielding finite-membered cardinal {} -element Ordinal-yielding increasing V71() decreasing non-decreasing non-increasing Cantor-normal-form () set
order_type_of a is epsilon-transitive epsilon-connected ordinal () set
RelIncl a is Relation-like well_founded well-ordering V34() V37() V39() V41() finite set
({}) is epsilon-transitive epsilon-connected ordinal () set
On {} is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty finite finite-yielding finite-membered cardinal {} -element Ordinal-yielding increasing V71() decreasing non-decreasing non-increasing Cantor-normal-form () set
RelIncl (On {}) is Relation-like well_founded well-ordering V34() V37() V39() V41() finite set
order_type_of (RelIncl (On {})) is epsilon-transitive epsilon-connected ordinal () set
RelIncl {} is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional well_founded well-ordering epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty V34() V37() V39() V41() finite finite-yielding finite-membered cardinal {} -element Ordinal-yielding increasing V71() decreasing non-decreasing non-increasing Cantor-normal-form () set
order_type_of (RelIncl {}) is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty finite finite-yielding finite-membered cardinal {} -element Ordinal-yielding increasing V71() decreasing non-decreasing non-increasing Cantor-normal-form () set
a is epsilon-transitive epsilon-connected ordinal () set
{a} is non empty trivial finite 1 -element set
({a}) is epsilon-transitive epsilon-connected ordinal () set
On {a} is () set
RelIncl (On {a}) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On {a})) is epsilon-transitive epsilon-connected ordinal () set
succ a is epsilon-transitive epsilon-connected ordinal non empty () set
a \/ {a} is non empty set
RelIncl {a} is Relation-like non empty V34() V37() V41() finite set
order_type_of (RelIncl {a}) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
{a,U} is non empty finite set
({a,U}) is epsilon-transitive epsilon-connected ordinal () set
On {a,U} is () set
RelIncl (On {a,U}) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On {a,U})) is epsilon-transitive epsilon-connected ordinal () set
card {a,U} is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
a \/ U is epsilon-transitive epsilon-connected ordinal () set
succ (a \/ U) is epsilon-transitive epsilon-connected ordinal non empty () set
{(a \/ U)} is non empty trivial finite 1 -element set
(a \/ U) \/ {(a \/ U)} is non empty set
a is epsilon-transitive epsilon-connected ordinal () set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl a is Relation-like well_founded well-ordering V34() V37() V39() V41() set
order_type_of (RelIncl a) is epsilon-transitive epsilon-connected ordinal () set
a is set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (a)),(RelIncl (On a))) is Relation-like Function-like set
f is epsilon-transitive epsilon-connected ordinal () set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom a is epsilon-transitive epsilon-connected ordinal () set
rng a is () set
a is set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (a)),(RelIncl (On a))) is Relation-like Function-like set
dom (a) is epsilon-transitive epsilon-connected ordinal () set
rng (a) is () set
f is epsilon-transitive epsilon-connected ordinal () set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom a is epsilon-transitive epsilon-connected ordinal () set
rng a is () set
a is () set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl a is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl a) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (a)),(RelIncl (On a))) is Relation-like Function-like set
rng (a) is () set
a is set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (a)),(RelIncl (On a))) is Relation-like Function-like set
dom (a) is epsilon-transitive epsilon-connected ordinal () set
card (dom (a)) is epsilon-transitive epsilon-connected ordinal cardinal () set
card (On a) is epsilon-transitive epsilon-connected ordinal cardinal () set
U is epsilon-transitive epsilon-connected ordinal () set
a is set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
canonical_isomorphism_of ((RelIncl (a)),(RelIncl (On a))) is Relation-like Function-like set
a is set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (a)),(RelIncl (On a))) is Relation-like Function-like set
U is set
g is set
o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom o is epsilon-transitive epsilon-connected ordinal () set
o . U is set
o . g is set
a is set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (a)),(RelIncl (On a))) is Relation-like Function-like set
U is set
g is set
o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom o is epsilon-transitive epsilon-connected ordinal () set
o . U is set
o . g is set
a is set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (a)),(RelIncl (On a))) is Relation-like Function-like set
f is epsilon-transitive epsilon-connected ordinal () set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom a is epsilon-transitive epsilon-connected ordinal () set
rng a is () set
a is () set
U is () set
a \/ U is set
g is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
g \/ o is epsilon-transitive epsilon-connected ordinal () set
a is () set
U is set
a \ U is Element of bool a
bool a is non empty set
g is epsilon-transitive epsilon-connected ordinal () set
a is () set
U is () set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl a is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl a) is epsilon-transitive epsilon-connected ordinal () set
(U) is epsilon-transitive epsilon-connected ordinal () set
On U is () set
RelIncl (On U) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On U)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl U is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl U) is epsilon-transitive epsilon-connected ordinal () set
(a) +^ (U) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ((a) +^ (U)) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
a \/ U is () set
RelIncl (a \/ U) is Relation-like well-ordering V34() V37() V41() set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
RelIncl (a) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (a)),(RelIncl (On a))) is Relation-like Function-like set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
RelIncl (U) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (U)),(RelIncl (On U))) is Relation-like Function-like set
(a) ^ (U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((a) ^ (U)) is epsilon-transitive epsilon-connected ordinal () set
dom (a) is epsilon-transitive epsilon-connected ordinal () set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (dom (U)) is epsilon-transitive epsilon-connected ordinal () set
rng (a) is () set
rng (U) is () set
field (RelIncl ((a) +^ (U))) is set
rng ((a) ^ (U)) is () set
field (RelIncl (a \/ U)) is set
(rng (a)) \/ (rng (U)) is () set
h is set
((a) ^ (U)) . h is set
f2 is set
((a) ^ (U)) . f2 is set
(a) . h is set
(a) . f2 is set
f1 is epsilon-transitive epsilon-connected ordinal () set
f1 -^ (dom (a)) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (f1 -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
(a) . h is set
(U) . (f1 -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
y -^ (dom (a)) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (y -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
(a) . f2 is set
(U) . (y -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
y -^ (dom (a)) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (y -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
f1 is epsilon-transitive epsilon-connected ordinal () set
f1 -^ (dom (a)) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (f1 -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
(U) . (f1 -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
(U) . (y -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
h is set
((a) ^ (U)) . h is set
f2 is set
[h,f2] is set
{h,f2} is non empty finite set
{h} is non empty trivial finite 1 -element set
{{h,f2},{h}} is non empty finite finite-membered set
((a) ^ (U)) . f2 is set
[(((a) ^ (U)) . h),(((a) ^ (U)) . f2)] is set
{(((a) ^ (U)) . h),(((a) ^ (U)) . f2)} is non empty finite set
{(((a) ^ (U)) . h)} is non empty trivial finite 1 -element set
{{(((a) ^ (U)) . h),(((a) ^ (U)) . f2)},{(((a) ^ (U)) . h)}} is non empty finite finite-membered set
y is epsilon-transitive epsilon-connected ordinal () set
f1 is epsilon-transitive epsilon-connected ordinal () set
[y,f1] is set
{y,f1} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,f1},{y}} is non empty finite finite-membered set
((a) ^ (U)) . y is epsilon-transitive epsilon-connected ordinal () set
(a) . y is epsilon-transitive epsilon-connected ordinal () set
((a) ^ (U)) . f1 is epsilon-transitive epsilon-connected ordinal () set
(a) . f1 is epsilon-transitive epsilon-connected ordinal () set
[(((a) ^ (U)) . y),(((a) ^ (U)) . f1)] is set
{(((a) ^ (U)) . y),(((a) ^ (U)) . f1)} is non empty finite set
{(((a) ^ (U)) . y)} is non empty trivial finite 1 -element set
{{(((a) ^ (U)) . y),(((a) ^ (U)) . f1)},{(((a) ^ (U)) . y)}} is non empty finite finite-membered set
field (RelIncl a) is set
y is epsilon-transitive epsilon-connected ordinal () set
f1 is epsilon-transitive epsilon-connected ordinal () set
f1 -^ (dom (a)) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (f1 -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
((a) ^ (U)) . y is epsilon-transitive epsilon-connected ordinal () set
(a) . y is epsilon-transitive epsilon-connected ordinal () set
((a) ^ (U)) . f1 is epsilon-transitive epsilon-connected ordinal () set
(U) . (f1 -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
f1 is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
f1 is epsilon-transitive epsilon-connected ordinal () set
y -^ (dom (a)) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (y -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
f1 -^ (dom (a)) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (f1 -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
y -^ (a) is epsilon-transitive epsilon-connected ordinal () set
f1 -^ (a) is epsilon-transitive epsilon-connected ordinal () set
[(y -^ (a)),(f1 -^ (a))] is set
{(y -^ (a)),(f1 -^ (a))} is non empty finite set
{(y -^ (a))} is non empty trivial finite 1 -element set
{{(y -^ (a)),(f1 -^ (a))},{(y -^ (a))}} is non empty finite finite-membered set
((a) ^ (U)) . f1 is epsilon-transitive epsilon-connected ordinal () set
(U) . (f1 -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
((a) ^ (U)) . y is epsilon-transitive epsilon-connected ordinal () set
(U) . (y -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
[(((a) ^ (U)) . y),(((a) ^ (U)) . f1)] is set
{(((a) ^ (U)) . y),(((a) ^ (U)) . f1)} is non empty finite set
{(((a) ^ (U)) . y)} is non empty trivial finite 1 -element set
{{(((a) ^ (U)) . y),(((a) ^ (U)) . f1)},{(((a) ^ (U)) . y)}} is non empty finite finite-membered set
field (RelIncl U) is set
y is epsilon-transitive epsilon-connected ordinal () set
f1 is epsilon-transitive epsilon-connected ordinal () set
field (RelIncl (a)) is set
field (RelIncl (U)) is set
y is epsilon-transitive epsilon-connected ordinal () set
f1 is epsilon-transitive epsilon-connected ordinal () set
((a) ^ (U)) . y is epsilon-transitive epsilon-connected ordinal () set
(a) . y is epsilon-transitive epsilon-connected ordinal () set
((a) ^ (U)) . f1 is epsilon-transitive epsilon-connected ordinal () set
(a) . f1 is epsilon-transitive epsilon-connected ordinal () set
[((a) . y),((a) . f1)] is set
{((a) . y),((a) . f1)} is non empty finite set
{((a) . y)} is non empty trivial finite 1 -element set
{{((a) . y),((a) . f1)},{((a) . y)}} is non empty finite finite-membered set
[y,f1] is set
{y,f1} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,f1},{y}} is non empty finite finite-membered set
y is epsilon-transitive epsilon-connected ordinal () set
f1 is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
f1 is epsilon-transitive epsilon-connected ordinal () set
((a) ^ (U)) . f1 is epsilon-transitive epsilon-connected ordinal () set
(a) . f1 is epsilon-transitive epsilon-connected ordinal () set
y -^ (dom (a)) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (y -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
((a) ^ (U)) . y is epsilon-transitive epsilon-connected ordinal () set
(U) . (y -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
f1 is epsilon-transitive epsilon-connected ordinal () set
y -^ (dom (a)) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (y -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
f1 -^ (dom (a)) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (f1 -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
(U) . (f1 -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
(U) . (y -^ (dom (a))) is epsilon-transitive epsilon-connected ordinal () set
((a) ^ (U)) . f1 is epsilon-transitive epsilon-connected ordinal () set
((a) ^ (U)) . y is epsilon-transitive epsilon-connected ordinal () set
[((U) . (y -^ (dom (a)))),((U) . (f1 -^ (dom (a))))] is set
{((U) . (y -^ (dom (a)))),((U) . (f1 -^ (dom (a))))} is non empty finite set
{((U) . (y -^ (dom (a))))} is non empty trivial finite 1 -element set
{{((U) . (y -^ (dom (a)))),((U) . (f1 -^ (dom (a))))},{((U) . (y -^ (dom (a))))}} is non empty finite finite-membered set
[(y -^ (dom (a))),(f1 -^ (dom (a)))] is set
{(y -^ (dom (a))),(f1 -^ (dom (a)))} is non empty finite set
{(y -^ (dom (a)))} is non empty trivial finite 1 -element set
{{(y -^ (dom (a))),(f1 -^ (dom (a)))},{(y -^ (dom (a)))}} is non empty finite finite-membered set
y is epsilon-transitive epsilon-connected ordinal () set
f1 is epsilon-transitive epsilon-connected ordinal () set
a is () set
U is () set
a \/ U is () set
((a \/ U)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
((a \/ U)) is epsilon-transitive epsilon-connected ordinal () set
On (a \/ U) is () set
RelIncl (On (a \/ U)) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On (a \/ U))) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a \/ U) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (a \/ U)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ((a \/ U)) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ((a \/ U))),(RelIncl (On (a \/ U)))) is Relation-like Function-like set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl a is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl a) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (a)),(RelIncl (On a))) is Relation-like Function-like set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
(U) is epsilon-transitive epsilon-connected ordinal () set
On U is () set
RelIncl (On U) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On U)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl U is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl U) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (U) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (U)),(RelIncl (On U))) is Relation-like Function-like set
(a) ^ (U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) +^ (U) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ((a) +^ (U)) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
a is () set
U is () set
a \/ U is () set
((a \/ U)) is epsilon-transitive epsilon-connected ordinal () set
On (a \/ U) is () set
RelIncl (On (a \/ U)) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On (a \/ U))) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a \/ U) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (a \/ U)) is epsilon-transitive epsilon-connected ordinal () set
(a) is epsilon-transitive epsilon-connected ordinal () set
On a is () set
RelIncl (On a) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On a)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl a is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl a) is epsilon-transitive epsilon-connected ordinal () set
(U) is epsilon-transitive epsilon-connected ordinal () set
On U is () set
RelIncl (On U) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On U)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl U is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl U) is epsilon-transitive epsilon-connected ordinal () set
(a) +^ (U) is epsilon-transitive epsilon-connected ordinal () set
((a \/ U)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
RelIncl ((a \/ U)) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ((a \/ U))),(RelIncl (On (a \/ U)))) is Relation-like Function-like set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
RelIncl (a) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (a)),(RelIncl (On a))) is Relation-like Function-like set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
RelIncl (U) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl (U)),(RelIncl (On U))) is Relation-like Function-like set
(a) ^ (U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((a \/ U)) is epsilon-transitive epsilon-connected ordinal () set
dom (a) is epsilon-transitive epsilon-connected ordinal () set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
(dom (a)) +^ (dom (U)) is epsilon-transitive epsilon-connected ordinal () set
(a) +^ (dom (U)) is epsilon-transitive epsilon-connected ordinal () set
a is set
U is Relation-like Function-like set
rng U is set
dom U is set
U . a is set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom a is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ))) is Relation-like Function-like set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom a is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is () set
g is set
o is epsilon-transitive epsilon-connected ordinal () Element of dom a
a is set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom U is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ))) is Relation-like Function-like set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
(U) . a is set
rng (U) is () set
a is epsilon-transitive epsilon-connected ordinal () Element of dom U
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom a is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ))) is Relation-like Function-like set
rng (a) is () set
rng a is () set
g is set
o is epsilon-transitive epsilon-connected ordinal () Element of dom a
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom a is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ))) is Relation-like Function-like set
a is set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom U is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ))) is Relation-like Function-like set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
(U) . a is set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom a is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ))) is Relation-like Function-like set
dom (a) is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
(a) . U is epsilon-transitive epsilon-connected ordinal () set
rng (a) is () set
o is epsilon-transitive epsilon-connected ordinal () Element of dom a
a . o is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom U is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ))) is Relation-like Function-like set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
rng (U) is () set
o is set
(U) . o is set
a is epsilon-transitive epsilon-connected ordinal () set
succ a is epsilon-transitive epsilon-connected ordinal non empty () set
{a} is non empty trivial finite 1 -element set
a \/ {a} is non empty set
U is epsilon-transitive epsilon-connected ordinal () set
g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(g) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom g is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ))) is Relation-like Function-like set
dom (g) is epsilon-transitive epsilon-connected ordinal () set
(g) . a is epsilon-transitive epsilon-connected ordinal () set
f is epsilon-transitive epsilon-connected ordinal () set
(g) . f is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
succ a is epsilon-transitive epsilon-connected ordinal non empty () set
{a} is non empty trivial finite 1 -element set
a \/ {a} is non empty set
U is epsilon-transitive epsilon-connected ordinal () set
g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(g) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom g is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ))) is Relation-like Function-like set
dom (g) is epsilon-transitive epsilon-connected ordinal () set
(g) . a is epsilon-transitive epsilon-connected ordinal () set
(g) . (succ a) is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
(g) . b is epsilon-transitive epsilon-connected ordinal () set
RelIncl { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } is Relation-like V34() V37() V41() set
a is set
union a is set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom U is epsilon-transitive epsilon-connected ordinal () set
U . (union a) is set
o is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
f is set
a is set
f is set
U . f is set
the Element of a is Element of a
f is set
a is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
y is set
c is set
x is epsilon-transitive epsilon-connected ordinal () set
succ b is epsilon-transitive epsilon-connected ordinal non empty () set
{b} is non empty trivial finite 1 -element set
b \/ {b} is non empty set
b is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite () set
U | b is Relation-like rng U -valued Function-like T-Sequence-like Ordinal-yielding set
rng U is () set
y is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom y is epsilon-transitive epsilon-connected ordinal () set
Union y is epsilon-transitive epsilon-connected ordinal () set
rng y is () set
union (rng y) is epsilon-transitive epsilon-connected ordinal () set
U . b is epsilon-transitive epsilon-connected ordinal () set
lim y is epsilon-transitive epsilon-connected ordinal () set
c is set
x is set
y . x is set
y is set
z is set
U . z is set
h is epsilon-transitive epsilon-connected ordinal () set
U . h is epsilon-transitive epsilon-connected ordinal () set
y . h is epsilon-transitive epsilon-connected ordinal () set
f2 is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
a is set
union a is set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom U is epsilon-transitive epsilon-connected ordinal () set
U . (union a) is set
g is set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite () set
g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(g) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom g is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ))) is Relation-like Function-like set
dom (g) is epsilon-transitive epsilon-connected ordinal () set
f is epsilon-transitive epsilon-connected ordinal () set
(g) . f is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
(g) . a is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite () set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom U is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ))) is Relation-like Function-like set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
(U) . a is epsilon-transitive epsilon-connected ordinal () set
(U) | a is Relation-like rng (U) -valued Function-like T-Sequence-like Ordinal-yielding set
rng (U) is () set
Union ((U) | a) is epsilon-transitive epsilon-connected ordinal () set
rng ((U) | a) is () set
union (rng ((U) | a)) is epsilon-transitive epsilon-connected ordinal () set
o is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
rng o is () set
U . ((U) . a) is epsilon-transitive epsilon-connected ordinal () set
dom o is epsilon-transitive epsilon-connected ordinal () set
a is set
b is set
o . b is set
(U) . b is set
union (rng o) is epsilon-transitive epsilon-connected ordinal () set
b is set
y is set
o . y is set
(U) . y is set
a is epsilon-transitive epsilon-connected ordinal () set
U . a is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
(U) . b is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
succ y is epsilon-transitive epsilon-connected ordinal non empty () set
{y} is non empty trivial finite 1 -element set
y \/ {y} is non empty set
(U) . y is epsilon-transitive epsilon-connected ordinal () set
o . y is epsilon-transitive epsilon-connected ordinal () set
(U) . (succ y) is epsilon-transitive epsilon-connected ordinal () set
o . (succ y) is epsilon-transitive epsilon-connected ordinal () set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing continuous non-decreasing () set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom a is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ))) is Relation-like Function-like set
g is epsilon-transitive epsilon-connected ordinal () set
dom (a) is epsilon-transitive epsilon-connected ordinal () set
(a) . g is epsilon-transitive epsilon-connected ordinal () set
(a) | g is Relation-like rng (a) -valued Function-like T-Sequence-like Ordinal-yielding set
rng (a) is () set
o is epsilon-transitive epsilon-connected ordinal () set
f is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
Union f is epsilon-transitive epsilon-connected ordinal () set
rng f is () set
union (rng f) is epsilon-transitive epsilon-connected ordinal () set
dom f is epsilon-transitive epsilon-connected ordinal () set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom a is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ))) is Relation-like Function-like set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom U is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ))) is Relation-like Function-like set
a is () set
f is () set
a \ f is () Element of bool a
bool a is non empty set
y is set
x is epsilon-transitive epsilon-connected ordinal () Element of dom a
c is set
y is epsilon-transitive epsilon-connected ordinal () Element of dom U
a . y is epsilon-transitive epsilon-connected ordinal () set
U . y is epsilon-transitive epsilon-connected ordinal () set
y is set
c is epsilon-transitive epsilon-connected ordinal () Element of dom a
a . c is epsilon-transitive epsilon-connected ordinal () set
U . c is epsilon-transitive epsilon-connected ordinal () set
f \/ (a \ f) is () set
((a \ f)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
((a \ f)) is epsilon-transitive epsilon-connected ordinal () set
On (a \ f) is () set
RelIncl (On (a \ f)) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On (a \ f))) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (a \ f) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (a \ f)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ((a \ f)) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ((a \ f))),(RelIncl (On (a \ f)))) is Relation-like Function-like set
(a) ^ ((a \ f)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is epsilon-transitive non empty subset-closed Tarski universal set
On a is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On a),(On a):] is Relation-like non empty set
bool [:(On a),(On a):] is non empty set
id (On a) is Relation-like On a -defined On a -valued Function-like one-to-one T-Sequence-like non empty V26( On a) V30( On a, On a) V30( On a, On a) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On a),(On a):]
U is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding Element of bool [:(On a),(On a):]
U is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive non empty subset-closed Tarski universal set
On a is epsilon-transitive epsilon-connected ordinal non empty () set
[:U,(On a):] is Relation-like set
bool [:U,(On a):] is non empty set
g is Relation-like U -defined On a -valued Function-like V26(U) V30(U, On a) Element of bool [:U,(On a):]
dom g is set
rng g is set
U is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive non empty subset-closed Tarski universal set
On a is epsilon-transitive epsilon-connected ordinal non empty () set
[:U,(On a):] is Relation-like set
bool [:U,(On a):] is non empty set
g is Relation-like U -defined On a -valued Function-like T-Sequence-like V26(U) V30(U, On a) Ordinal-yielding Element of bool [:U,(On a):]
o is set
g . o is set
dom g is epsilon-transitive epsilon-connected ordinal () set
dom g is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive non empty subset-closed Tarski universal set
On U is epsilon-transitive epsilon-connected ordinal non empty () set
[:a,(On U):] is Relation-like set
bool [:a,(On U):] is non empty set
g is Relation-like a -defined On U -valued Function-like T-Sequence-like V26(a) V30(a, On U) Ordinal-yielding Element of bool [:a,(On U):]
Union g is epsilon-transitive epsilon-connected ordinal () set
rng g is () set
union (rng g) is epsilon-transitive epsilon-connected ordinal () set
dom g is epsilon-transitive epsilon-connected ordinal () set
card (dom g) is epsilon-transitive epsilon-connected ordinal cardinal () set
card U is epsilon-transitive epsilon-connected ordinal non empty cardinal () set
card (rng g) is epsilon-transitive epsilon-connected ordinal cardinal () set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive non empty subset-closed Tarski universal set
On U is epsilon-transitive epsilon-connected ordinal non empty () set
[:a,(On U):] is Relation-like set
bool [:a,(On U):] is non empty set
g is Relation-like a -defined On U -valued Function-like T-Sequence-like V26(a) V30(a, On U) Ordinal-yielding Element of bool [:a,(On U):]
sup g is epsilon-transitive epsilon-connected ordinal () set
rng g is () set
sup (rng g) is epsilon-transitive epsilon-connected ordinal () set
Union g is epsilon-transitive epsilon-connected ordinal () set
union (rng g) is epsilon-transitive epsilon-connected ordinal () set
On (rng g) is () set
o is epsilon-transitive epsilon-connected ordinal () Element of U
succ o is epsilon-transitive epsilon-connected ordinal non empty () Element of U
{o} is non empty trivial finite 1 -element set
o \/ {o} is non empty set
F₁() is epsilon-transitive non empty subset-closed Tarski universal set
On F₁() is epsilon-transitive epsilon-connected ordinal non empty () set
[:omega,(On F₁()):] is Relation-like non empty non trivial non finite set
bool [:omega,(On F₁()):] is non empty non trivial non finite set
F₃() is Relation-like omega -defined On F₁() -valued Function-like T-Sequence-like non empty V26( omega ) V30( omega , On F₁()) Ordinal-yielding Element of bool [:omega,(On F₁()):]
(F₁(),omega,F₃(),0) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
F₂() is epsilon-transitive epsilon-connected ordinal () Element of F₁()
Union F₃() is epsilon-transitive epsilon-connected ordinal () set
rng F₃() is non empty () set
union (rng F₃()) is epsilon-transitive epsilon-connected ordinal () set
F₄((Union F₃())) is epsilon-transitive epsilon-connected ordinal () set
dom F₃() is epsilon-transitive epsilon-connected ordinal non empty () set
U is epsilon-transitive epsilon-connected ordinal () set
F₄(U) is epsilon-transitive epsilon-connected ordinal () set
F₄(F₄(U)) is epsilon-transitive epsilon-connected ordinal () set
F₄(F₂()) is epsilon-transitive epsilon-connected ordinal () set
U is set
(F₁(),omega,F₃(),U) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
o is epsilon-transitive epsilon-connected ordinal natural finite cardinal () set
(F₁(),omega,F₃(),o) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
succ o is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{o} is non empty trivial finite finite-membered 1 -element set
o \/ {o} is non empty finite set
(F₁(),omega,F₃(),(succ o)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
o + 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of NAT
(F₁(),omega,F₃(),(o + 1)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
g is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of omega
(F₁(),omega,F₃(),g) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
{F₂()} is non empty trivial finite 1 -element Element of F₁()
U is set
g is set
(F₁(),omega,F₃(),g) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
U is epsilon-transitive epsilon-connected ordinal () set
F₄(U) is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
succ U is epsilon-transitive epsilon-connected ordinal non empty () set
{U} is non empty trivial finite 1 -element set
U \/ {U} is non empty set
(F₁(),omega,F₃(),(succ U)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
(F₁(),omega,F₃(),U) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
F₄((F₁(),omega,F₃(),U)) is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
(F₁(),omega,F₃(),U) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
succ U is epsilon-transitive epsilon-connected ordinal non empty () set
{U} is non empty trivial finite 1 -element set
U \/ {U} is non empty set
(F₁(),omega,F₃(),(succ U)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
succ 0 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{0} is functional non empty trivial finite finite-membered 1 -element set
0 \/ {0} is non empty finite finite-membered set
(F₁(),omega,F₃(),(succ 0)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
o is epsilon-transitive epsilon-connected ordinal natural finite cardinal () set
(F₁(),omega,F₃(),o) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
succ o is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{o} is non empty trivial finite finite-membered 1 -element set
o \/ {o} is non empty finite set
(F₁(),omega,F₃(),(succ o)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
o + 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of NAT
(o + 1) + 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of NAT
succ (o + 1) is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{(o + 1)} is non empty trivial finite finite-membered 1 -element set
(o + 1) \/ {(o + 1)} is non empty finite set
(F₁(),omega,F₃(),(o + 1)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
F₄((F₁(),omega,F₃(),o)) is epsilon-transitive epsilon-connected ordinal () set
(F₁(),omega,F₃(),((o + 1) + 1)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
F₄((F₁(),omega,F₃(),(o + 1))) is epsilon-transitive epsilon-connected ordinal () set
(F₁(),omega,F₃(),(succ (o + 1))) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
g is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of omega
(F₁(),omega,F₃(),g) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
succ g is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{g} is non empty trivial finite finite-membered 1 -element set
g \/ {g} is non empty finite set
(F₁(),omega,F₃(),(succ g)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom U is epsilon-transitive epsilon-connected ordinal () set
succ 0 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{0} is functional non empty trivial finite finite-membered 1 -element set
0 \/ {0} is non empty finite finite-membered set
(F₁(),omega,F₃(),1) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
g is epsilon-transitive epsilon-connected ordinal () set
o is set
(F₁(),omega,F₃(),o) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
succ g is epsilon-transitive epsilon-connected ordinal non empty () set
{g} is non empty trivial finite 1 -element set
g \/ {g} is non empty set
f is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of omega
(F₁(),omega,F₃(),f) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
succ f is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{f} is non empty trivial finite finite-membered 1 -element set
f \/ {f} is non empty finite set
(F₁(),omega,F₃(),(succ f)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
g is epsilon-transitive epsilon-connected ordinal () set
U . g is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
U . o is epsilon-transitive epsilon-connected ordinal () set
F₄(g) is epsilon-transitive epsilon-connected ordinal () set
F₄(o) is epsilon-transitive epsilon-connected ordinal () set
sup U is epsilon-transitive epsilon-connected ordinal () set
rng U is () set
sup (rng U) is epsilon-transitive epsilon-connected ordinal () set
lim U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
f is epsilon-transitive epsilon-connected ordinal () set
a is set
U . a is set
b is epsilon-transitive epsilon-connected ordinal () set
y is set
(F₁(),omega,F₃(),y) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
c is epsilon-transitive epsilon-connected ordinal () set
(F₁(),omega,F₃(),c) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
succ c is epsilon-transitive epsilon-connected ordinal non empty () set
{c} is non empty trivial finite 1 -element set
c \/ {c} is non empty set
(F₁(),omega,F₃(),(succ c)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
F₄((F₁(),omega,F₃(),c)) is epsilon-transitive epsilon-connected ordinal () set
F₄(b) is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
F₄(g) is epsilon-transitive epsilon-connected ordinal () set
o is set
f is set
(F₁(),omega,F₃(),f) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
b is epsilon-transitive epsilon-connected ordinal natural finite cardinal () set
(F₁(),omega,F₃(),b) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
F₄((F₁(),omega,F₃(),b)) is epsilon-transitive epsilon-connected ordinal () set
succ b is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{b} is non empty trivial finite finite-membered 1 -element set
b \/ {b} is non empty finite set
(F₁(),omega,F₃(),(succ b)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
b + 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of NAT
(F₁(),omega,F₃(),(b + 1)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
a is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of omega
(F₁(),omega,F₃(),a) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
union g is epsilon-transitive epsilon-connected ordinal () set
F₁() is epsilon-transitive non empty subset-closed Tarski universal set
F₂() is epsilon-transitive epsilon-connected ordinal () Element of F₁()
succ F₂() is epsilon-transitive epsilon-connected ordinal non empty () Element of F₁()
{F₂()} is non empty trivial finite 1 -element set
F₂() \/ {F₂()} is non empty set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom a is epsilon-transitive epsilon-connected ordinal () set
a . {} is epsilon-transitive epsilon-connected ordinal () set
rng a is () set
On F₁() is epsilon-transitive epsilon-connected ordinal non empty () set
U is set
g is set
a . g is set
a . 0 is epsilon-transitive epsilon-connected ordinal () set
f is epsilon-transitive epsilon-connected ordinal natural finite cardinal () set
a . f is epsilon-transitive epsilon-connected ordinal () set
f + 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of NAT
a . (f + 1) is epsilon-transitive epsilon-connected ordinal () set
succ f is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{f} is non empty trivial finite finite-membered 1 -element set
f \/ {f} is non empty finite set
F₃((a . f)) is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of NAT
a . o is epsilon-transitive epsilon-connected ordinal () set
[:omega,(On F₁()):] is Relation-like non empty non trivial non finite set
bool [:omega,(On F₁()):] is non empty non trivial non finite set
g is epsilon-transitive epsilon-connected ordinal () set
F₃(g) is epsilon-transitive epsilon-connected ordinal () set
F₃(F₃(g)) is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
F₃(g) is epsilon-transitive epsilon-connected ordinal () set
U is Relation-like omega -defined On F₁() -valued Function-like T-Sequence-like non empty V26( omega ) V30( omega , On F₁()) Ordinal-yielding Element of bool [:omega,(On F₁()):]
g is epsilon-transitive epsilon-connected ordinal () set
(F₁(),omega,U,g) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
(F₁(),omega,U,0) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
f is epsilon-transitive epsilon-connected ordinal natural finite cardinal () set
(F₁(),omega,U,f) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
f + 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of NAT
succ f is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{f} is non empty trivial finite finite-membered 1 -element set
f \/ {f} is non empty finite set
(F₁(),omega,U,(f + 1)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
F₃((F₁(),omega,U,f)) is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal natural finite cardinal () Element of omega
(F₁(),omega,U,o) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
g is epsilon-transitive epsilon-connected ordinal () set
dom U is epsilon-transitive epsilon-connected ordinal non empty () set
U . g is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
U . o is epsilon-transitive epsilon-connected ordinal () set
(F₁(),omega,U,g) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
g +^ {} is epsilon-transitive epsilon-connected ordinal () set
(F₁(),omega,U,(g +^ {})) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
f is epsilon-transitive epsilon-connected ordinal () set
g +^ f is epsilon-transitive epsilon-connected ordinal () set
(F₁(),omega,U,(g +^ f)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
succ f is epsilon-transitive epsilon-connected ordinal non empty () set
{f} is non empty trivial finite 1 -element set
f \/ {f} is non empty set
g +^ (succ f) is epsilon-transitive epsilon-connected ordinal () set
(F₁(),omega,U,(g +^ (succ f))) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
succ (g +^ f) is epsilon-transitive epsilon-connected ordinal non empty () set
{(g +^ f)} is non empty trivial finite 1 -element set
(g +^ f) \/ {(g +^ f)} is non empty set
a is epsilon-transitive epsilon-connected ordinal () set
F₃(a) is epsilon-transitive epsilon-connected ordinal () set
f is epsilon-transitive epsilon-connected ordinal () set
g +^ f is epsilon-transitive epsilon-connected ordinal () set
(F₁(),omega,U,(g +^ f)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
f is epsilon-transitive epsilon-connected ordinal () set
g +^ f is epsilon-transitive epsilon-connected ordinal () set
sup U is epsilon-transitive epsilon-connected ordinal () set
rng U is non empty () set
sup (rng U) is epsilon-transitive epsilon-connected ordinal () set
g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom g is epsilon-transitive epsilon-connected ordinal () set
F₃((sup U)) is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
g . o is epsilon-transitive epsilon-connected ordinal () set
f is epsilon-transitive epsilon-connected ordinal () set
g . f is epsilon-transitive epsilon-connected ordinal () set
F₃(o) is epsilon-transitive epsilon-connected ordinal () set
F₃(f) is epsilon-transitive epsilon-connected ordinal () set
sup g is epsilon-transitive epsilon-connected ordinal () set
rng g is () set
sup (rng g) is epsilon-transitive epsilon-connected ordinal () set
lim g is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
f is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
b is set
g . b is set
y is epsilon-transitive epsilon-connected ordinal () set
c is epsilon-transitive epsilon-connected ordinal () set
dom U is epsilon-transitive epsilon-connected ordinal non empty () set
x is set
(F₁(),omega,U,x) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
y is epsilon-transitive epsilon-connected ordinal () set
succ y is epsilon-transitive epsilon-connected ordinal non empty () set
{y} is non empty trivial finite 1 -element set
y \/ {y} is non empty set
(F₁(),omega,U,(succ y)) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
F₃(c) is epsilon-transitive epsilon-connected ordinal () set
F₃(y) is epsilon-transitive epsilon-connected ordinal () set
(F₁(),omega,U,0) is epsilon-transitive epsilon-connected ordinal () Element of F₁()
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive non empty subset-closed Tarski universal set
On U is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On U),(On U):] is Relation-like non empty set
bool [:(On U),(On U):] is non empty set
g is Relation-like On U -defined On U -valued Function-like T-Sequence-like non empty V26( On U) V30( On U, On U) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On U),(On U):]
dom g is epsilon-transitive epsilon-connected ordinal non empty () set
f is epsilon-transitive epsilon-connected ordinal () set
(U,(On U),g,f) is epsilon-transitive epsilon-connected ordinal () Element of U
f is epsilon-transitive epsilon-connected ordinal () set
(U,(On U),g,f) is epsilon-transitive epsilon-connected ordinal () Element of U
a is epsilon-transitive epsilon-connected ordinal () set
(U,(On U),g,a) is epsilon-transitive epsilon-connected ordinal () Element of U
f is epsilon-transitive epsilon-connected ordinal () Element of U
(U,(On U),g,f) is epsilon-transitive epsilon-connected ordinal () Element of U
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom a is epsilon-transitive epsilon-connected ordinal () set
g | f is Relation-like On U -defined f -defined On U -defined On U -valued rng g -valued Function-like T-Sequence-like Ordinal-yielding Element of bool [:(On U),(On U):]
rng g is non empty () set
dom (g | f) is epsilon-transitive epsilon-connected ordinal () set
b is set
(g | f) . b is set
(U,(On U),g,b) is epsilon-transitive epsilon-connected ordinal () Element of U
a . b is set
o is epsilon-transitive epsilon-connected ordinal () Element of U
f is epsilon-transitive epsilon-connected ordinal () Element of U
(U,(On U),g,f) is epsilon-transitive epsilon-connected ordinal () Element of U
a is epsilon-transitive non empty subset-closed Tarski universal set
On a is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On a),(On a):] is Relation-like non empty set
bool [:(On a),(On a):] is non empty set
U is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On a),(On a):]
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing continuous non-decreasing () set
dom U is epsilon-transitive epsilon-connected ordinal non empty () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ is_a_fixpoint_of U } ))) is Relation-like Function-like set
rng U is non empty () set
rng (U) is () set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
0-element_of a is epsilon-transitive epsilon-connected ordinal () Element of a
f is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
(U) . a is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
succ b is epsilon-transitive epsilon-connected ordinal non empty () set
{b} is non empty trivial finite 1 -element set
b \/ {b} is non empty set
(U) . b is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
c is epsilon-transitive epsilon-connected ordinal () set
(U) . c is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
(U) .: b is set
card ((U) .: b) is epsilon-transitive epsilon-connected ordinal cardinal () set
card b is epsilon-transitive epsilon-connected ordinal cardinal () set
card a is epsilon-transitive epsilon-connected ordinal non empty cardinal () set
union ((U) .: b) is set
c is epsilon-transitive epsilon-connected ordinal () set
x is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
y is set
(U) . y is set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom a is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ is_a_fixpoint_of a } ))) is Relation-like Function-like set
dom (a) is epsilon-transitive epsilon-connected ordinal () set
a . {} is epsilon-transitive epsilon-connected ordinal () set
(a) . {} is epsilon-transitive epsilon-connected ordinal () set
(a) . 0 is epsilon-transitive epsilon-connected ordinal () set
a . ((a) . 0) is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
a . g is epsilon-transitive epsilon-connected ordinal () set
(a) . g is epsilon-transitive epsilon-connected ordinal () set
succ g is epsilon-transitive epsilon-connected ordinal non empty () set
{g} is non empty trivial finite 1 -element set
g \/ {g} is non empty set
a . (succ g) is epsilon-transitive epsilon-connected ordinal () set
(a) . (succ g) is epsilon-transitive epsilon-connected ordinal () set
a . ((a) . (succ g)) is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
a . g is epsilon-transitive epsilon-connected ordinal () set
(a) . g is epsilon-transitive epsilon-connected ordinal () set
a | g is Relation-like rng a -valued Function-like T-Sequence-like Ordinal-yielding set
rng a is () set
(a) | g is Relation-like rng (a) -valued Function-like T-Sequence-like Ordinal-yielding set
rng (a) is () set
lim (a | g) is epsilon-transitive epsilon-connected ordinal () set
lim ((a) | g) is epsilon-transitive epsilon-connected ordinal () set
dom (a | g) is epsilon-transitive epsilon-connected ordinal () set
dom ((a) | g) is epsilon-transitive epsilon-connected ordinal () set
Union (a | g) is epsilon-transitive epsilon-connected ordinal () set
rng (a | g) is () set
union (rng (a | g)) is epsilon-transitive epsilon-connected ordinal () set
Union ((a) | g) is epsilon-transitive epsilon-connected ordinal () set
rng ((a) | g) is () set
union (rng ((a) | g)) is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
f is set
(a | g) . f is set
a . f is set
((a) | g) . f is set
(a) . f is set
a is epsilon-transitive non empty subset-closed Tarski universal set
U is epsilon-transitive epsilon-connected ordinal () Element of a
g is epsilon-transitive epsilon-connected ordinal () Element of a
exp (U,g) is epsilon-transitive epsilon-connected ordinal () set
1-element_of a is epsilon-transitive epsilon-connected ordinal non empty () Element of a
exp (U,0) is epsilon-transitive epsilon-connected ordinal () set
succ 0 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{0} is functional non empty trivial finite finite-membered 1 -element set
0 \/ {0} is non empty finite finite-membered set
exp (U,{}) is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
exp (U,o) is epsilon-transitive epsilon-connected ordinal () set
succ o is epsilon-transitive epsilon-connected ordinal non empty () set
{o} is non empty trivial finite 1 -element set
o \/ {o} is non empty set
exp (U,(succ o)) is epsilon-transitive epsilon-connected ordinal () set
f is epsilon-transitive epsilon-connected ordinal () Element of a
exp (U,f) is epsilon-transitive epsilon-connected ordinal () set
succ f is epsilon-transitive epsilon-connected ordinal non empty () Element of a
{f} is non empty trivial finite 1 -element set
f \/ {f} is non empty set
exp (U,(succ f)) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () Element of a
U *^ a is epsilon-transitive epsilon-connected ordinal () Element of a
o is epsilon-transitive epsilon-connected ordinal () set
exp (U,o) is epsilon-transitive epsilon-connected ordinal () set
f is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom f is epsilon-transitive epsilon-connected ordinal () set
rng f is () set
On a is epsilon-transitive epsilon-connected ordinal non empty () set
a is set
b is set
f . b is set
y is epsilon-transitive epsilon-connected ordinal () set
exp (U,y) is epsilon-transitive epsilon-connected ordinal () set
[:o,(On a):] is Relation-like set
bool [:o,(On a):] is non empty set
a is Relation-like o -defined On a -valued Function-like T-Sequence-like V26(o) V30(o, On a) Ordinal-yielding Element of bool [:o,(On a):]
lim a is epsilon-transitive epsilon-connected ordinal () set
Union a is epsilon-transitive epsilon-connected ordinal () set
rng a is () set
union (rng a) is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive non empty subset-closed Tarski universal set
On a is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On a),(On a):] is Relation-like non empty set
bool [:(On a),(On a):] is non empty set
g is epsilon-transitive epsilon-connected ordinal () Element of a
o is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding Element of bool [:(On a),(On a):]
g is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding Element of bool [:(On a),(On a):]
o is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding Element of bool [:(On a),(On a):]
f is epsilon-transitive epsilon-connected ordinal () Element of On a
(a,(On a),g,f) is epsilon-transitive epsilon-connected ordinal () Element of a
exp (U,f) is epsilon-transitive epsilon-connected ordinal () set
(a,(On a),o,f) is epsilon-transitive epsilon-connected ordinal () Element of a
a is epsilon-transitive non empty subset-closed Tarski universal set
1-element_of a is epsilon-transitive epsilon-connected ordinal non empty () Element of a
succ (1-element_of a) is epsilon-transitive epsilon-connected ordinal non empty () Element of a
{(1-element_of a)} is non empty trivial finite 1 -element set
(1-element_of a) \/ {(1-element_of a)} is non empty set
a is epsilon-transitive non empty subset-closed Tarski universal set
U is epsilon-transitive epsilon-connected ordinal non trivial () Element of a
(a,U) is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding Element of bool [:(On a),(On a):]
On a is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On a),(On a):] is Relation-like non empty set
bool [:(On a),(On a):] is non empty set
dom (a,U) is epsilon-transitive epsilon-connected ordinal non empty () set
succ 0 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{0} is functional non empty trivial finite finite-membered 1 -element set
0 \/ {0} is non empty finite finite-membered set
o is epsilon-transitive epsilon-connected ordinal () set
(a,(On a),(a,U),o) is epsilon-transitive epsilon-connected ordinal () Element of a
exp (U,o) is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
(a,U) . o is epsilon-transitive epsilon-connected ordinal () set
(a,U) | o is Relation-like o -defined On a -defined On a -valued rng (a,U) -valued Function-like T-Sequence-like Ordinal-yielding set
rng (a,U) is non empty () set
f is epsilon-transitive epsilon-connected ordinal () set
(a,(On a),(a,U),o) is epsilon-transitive epsilon-connected ordinal () Element of a
(a,U) | o is Relation-like On a -defined o -defined On a -defined On a -valued rng (a,U) -valued Function-like T-Sequence-like Ordinal-yielding Element of bool [:(On a),(On a):]
dom ((a,U) | o) is epsilon-transitive epsilon-connected ordinal () set
exp (U,o) is epsilon-transitive epsilon-connected ordinal () set
Union ((a,U) | o) is epsilon-transitive epsilon-connected ordinal () set
rng ((a,U) | o) is () set
union (rng ((a,U) | o)) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
exp (U,b) is epsilon-transitive epsilon-connected ordinal () set
(a,(On a),(a,U),b) is epsilon-transitive epsilon-connected ordinal () Element of a
((a,U) | o) . b is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
b is set
((a,U) | o) . b is set
y is epsilon-transitive epsilon-connected ordinal () set
exp (U,y) is epsilon-transitive epsilon-connected ordinal () set
(a,(On a),(a,U),y) is epsilon-transitive epsilon-connected ordinal () Element of a
((a,U) | o) . y is epsilon-transitive epsilon-connected ordinal () set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
<%a%> is Relation-like NAT -defined Function-like constant T-Sequence-like non empty trivial finite 1 -element V71() set
U is set
rng <%a%> is non empty trivial finite 1 -element set
{a} is functional non empty trivial finite 1 -element set
a is Relation-like Function-like set
{a} is functional non empty trivial finite 1 -element set
U is Relation-like Function-like set
g is Relation-like Function-like set
dom U is set
dom g is set
a is Relation-like Function-like set
<%a%> is Relation-like NAT -defined Function-like constant T-Sequence-like non empty trivial finite 1 -element V71() set
rng <%a%> is non empty trivial finite 1 -element set
{a} is functional non empty trivial finite 1 -element with_common_domain set
the Relation-like Function-like T-Sequence-like Ordinal-yielding set is Relation-like Function-like T-Sequence-like Ordinal-yielding set
<% the Relation-like Function-like T-Sequence-like Ordinal-yielding set %> is Relation-like NAT -defined Function-like constant T-Sequence-like non empty trivial finite 1 -element V71() () () set
a is Relation-like Function-like () set
U is set
a . U is set
dom a is set
rng a is set
dom a is set
a is Relation-like Function-like () set
U is set
a . U is Relation-like Function-like set
dom a is set
rng a is set
dom a is set
a is Relation-like Function-like T-Sequence-like set
U is epsilon-transitive epsilon-connected ordinal () set
a | U is Relation-like rng a -valued Function-like T-Sequence-like set
rng a is set
a is Relation-like Function-like () set
U is set
a | U is Relation-like Function-like set
g is set
rng (a | U) is set
rng a is set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
[:a,U:] is Relation-like set
bool [:a,U:] is non empty set
g is Relation-like a -defined U -valued Function-like V30(a,U) Element of bool [:a,U:]
rng g is set
dom g is set
a is Relation-like Function-like T-Sequence-like () set
a . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (a . 0) is epsilon-transitive epsilon-connected ordinal () set
rng a is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
a is Relation-like Function-like T-Sequence-like () set
a . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (a . 0) is epsilon-transitive epsilon-connected ordinal () set
rng a is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
U is set
g is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0)
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
g is Relation-like Function-like T-Sequence-like () set
dom g is epsilon-transitive epsilon-connected ordinal () set
(g) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (g . 0) is epsilon-transitive epsilon-connected ordinal () set
rng g is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom (g) is epsilon-transitive epsilon-connected ordinal () set
(g) . U is epsilon-transitive epsilon-connected ordinal () set
g . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
rng (g) is () set
a is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0)
a is set
U is Relation-like Function-like T-Sequence-like () set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (U . 0) is epsilon-transitive epsilon-connected ordinal () set
rng U is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
(U) . a is set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom a is epsilon-transitive epsilon-connected ordinal () set
U is Relation-like Function-like T-Sequence-like () set
rng U is set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (U . 0) is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
(U) . g is epsilon-transitive epsilon-connected ordinal () set
rng (U) is () set
f is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0)
a . f is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
U is Relation-like Function-like T-Sequence-like () set
dom U is epsilon-transitive epsilon-connected ordinal () set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (U . 0) is epsilon-transitive epsilon-connected ordinal () set
rng U is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
f is set
U . f is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g is () set
rng (U) is () set
o is set
(U) . o is set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite () set
g is Relation-like Function-like T-Sequence-like () set
dom g is epsilon-transitive epsilon-connected ordinal () set
(g) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (g . 0) is epsilon-transitive epsilon-connected ordinal () set
rng g is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . 0) : ( b₁ in dom (g . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng g or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom (g) is epsilon-transitive epsilon-connected ordinal () set
f is epsilon-transitive epsilon-connected ordinal () set
g . f is Relation-like Function-like T-Sequence-like Ordinal-yielding set
f is epsilon-transitive epsilon-connected ordinal () set
(g) . f is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
(g) . a is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite () set
U is Relation-like Function-like T-Sequence-like () set
dom U is epsilon-transitive epsilon-connected ordinal () set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (U . 0) is epsilon-transitive epsilon-connected ordinal () set
rng U is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
(U) . a is epsilon-transitive epsilon-connected ordinal () set
(U) | a is Relation-like rng (U) -valued Function-like T-Sequence-like Ordinal-yielding set
rng (U) is () set
Union ((U) | a) is epsilon-transitive epsilon-connected ordinal () set
rng ((U) | a) is () set
union (rng ((U) | a)) is epsilon-transitive epsilon-connected ordinal () set
o is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
rng o is () set
a is epsilon-transitive epsilon-connected ordinal () set
U . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (U . a) is epsilon-transitive epsilon-connected ordinal () set
(U . a) . ((U) . a) is epsilon-transitive epsilon-connected ordinal () set
dom o is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
U . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
b is set
y is set
o . y is set
(U) . y is set
union (rng o) is epsilon-transitive epsilon-connected ordinal () set
b is set
y is set
o . y is set
(U) . y is set
a is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
U . b is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (U . b) is epsilon-transitive epsilon-connected ordinal () set
c is set
(U . b) . a is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
(U) . b is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
succ y is epsilon-transitive epsilon-connected ordinal non empty () set
{y} is non empty trivial finite 1 -element set
y \/ {y} is non empty set
(U) . y is epsilon-transitive epsilon-connected ordinal () set
o . y is epsilon-transitive epsilon-connected ordinal () set
(U) . (succ y) is epsilon-transitive epsilon-connected ordinal () set
o . (succ y) is epsilon-transitive epsilon-connected ordinal () set
a is Relation-like Function-like T-Sequence-like () set
dom a is epsilon-transitive epsilon-connected ordinal () set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (a . 0) is epsilon-transitive epsilon-connected ordinal () set
rng a is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
g is epsilon-transitive epsilon-connected ordinal () set
dom (a) is epsilon-transitive epsilon-connected ordinal () set
(a) . g is epsilon-transitive epsilon-connected ordinal () set
(a) | g is Relation-like rng (a) -valued Function-like T-Sequence-like Ordinal-yielding set
rng (a) is () set
o is epsilon-transitive epsilon-connected ordinal () set
f is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
Union f is epsilon-transitive epsilon-connected ordinal () set
rng f is () set
union (rng f) is epsilon-transitive epsilon-connected ordinal () set
dom f is epsilon-transitive epsilon-connected ordinal () set
a is Relation-like Function-like T-Sequence-like () set
dom a is epsilon-transitive epsilon-connected ordinal () set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (a . 0) is epsilon-transitive epsilon-connected ordinal () set
rng a is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom (a) is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
(a) . U is epsilon-transitive epsilon-connected ordinal () set
g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g . U is epsilon-transitive epsilon-connected ordinal () set
o is set
a . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom g is epsilon-transitive epsilon-connected ordinal () set
g . {} is epsilon-transitive epsilon-connected ordinal () set
(a) . {} is epsilon-transitive epsilon-connected ordinal () set
(a) . 0 is epsilon-transitive epsilon-connected ordinal () set
g . ((a) . 0) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
g . a is epsilon-transitive epsilon-connected ordinal () set
(a) . a is epsilon-transitive epsilon-connected ordinal () set
succ a is epsilon-transitive epsilon-connected ordinal non empty () set
{a} is non empty trivial finite 1 -element set
a \/ {a} is non empty set
g . (succ a) is epsilon-transitive epsilon-connected ordinal () set
(a) . (succ a) is epsilon-transitive epsilon-connected ordinal () set
g . ((a) . (succ a)) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
g . a is epsilon-transitive epsilon-connected ordinal () set
(a) . a is epsilon-transitive epsilon-connected ordinal () set
g | a is Relation-like rng g -valued Function-like T-Sequence-like Ordinal-yielding set
rng g is () set
(a) | a is Relation-like rng (a) -valued Function-like T-Sequence-like Ordinal-yielding set
rng (a) is () set
lim (g | a) is epsilon-transitive epsilon-connected ordinal () set
lim ((a) | a) is epsilon-transitive epsilon-connected ordinal () set
dom (g | a) is epsilon-transitive epsilon-connected ordinal () set
dom ((a) | a) is epsilon-transitive epsilon-connected ordinal () set
Union (g | a) is epsilon-transitive epsilon-connected ordinal () set
rng (g | a) is () set
union (rng (g | a)) is epsilon-transitive epsilon-connected ordinal () set
Union ((a) | a) is epsilon-transitive epsilon-connected ordinal () set
rng ((a) | a) is () set
union (rng ((a) | a)) is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
y is set
(g | a) . y is set
g . y is set
((a) | a) . y is set
(a) . y is set
a is Relation-like Function-like T-Sequence-like () set
dom a is epsilon-transitive epsilon-connected ordinal () set
U is Relation-like Function-like T-Sequence-like () set
dom U is epsilon-transitive epsilon-connected ordinal () set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (a . 0) is epsilon-transitive epsilon-connected ordinal () set
rng a is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0) : ( b₁ in dom (a . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng a or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (U . 0) is epsilon-transitive epsilon-connected ordinal () set
rng U is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0) : ( b₁ in dom (U . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng U or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
y is () set
b is () set
y \ b is () Element of bool y
bool y is non empty set
x is set
z is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0)
y is set
h is epsilon-transitive epsilon-connected ordinal () Element of dom (U . 0)
f2 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
y is set
a . y is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U . y is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom f2 is epsilon-transitive epsilon-connected ordinal () set
f2 . h is epsilon-transitive epsilon-connected ordinal () set
(U . y) . h is epsilon-transitive epsilon-connected ordinal () set
x is set
y is epsilon-transitive epsilon-connected ordinal () Element of dom (a . 0)
z is Relation-like Function-like T-Sequence-like Ordinal-yielding set
h is set
U . h is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a . h is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (a . h) is epsilon-transitive epsilon-connected ordinal () set
dom z is epsilon-transitive epsilon-connected ordinal () set
(a . h) . y is epsilon-transitive epsilon-connected ordinal () set
z . y is epsilon-transitive epsilon-connected ordinal () set
b \/ (y \ b) is () set
((y \ b)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
((y \ b)) is epsilon-transitive epsilon-connected ordinal () set
On (y \ b) is () set
RelIncl (On (y \ b)) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On (y \ b))) is epsilon-transitive epsilon-connected ordinal () set
RelIncl (y \ b) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (y \ b)) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ((y \ b)) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ((y \ b))),(RelIncl (On (y \ b)))) is Relation-like Function-like set
(a) ^ ((y \ b)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is Relation-like Function-like T-Sequence-like () set
a . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (a . 0) is epsilon-transitive epsilon-connected ordinal () set
dom a is epsilon-transitive epsilon-connected ordinal () set
{ ((a . b₁) . a₁) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ in dom a } is set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
U . g is epsilon-transitive epsilon-connected ordinal () set
{ ((a . b₁) . g) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ in dom a } is set
union { ((a . b₁) . g) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ in dom a } is set
On H₁(g) is () set
union (On H₁(g)) is epsilon-transitive epsilon-connected ordinal () set
o is set
f is epsilon-transitive epsilon-connected ordinal () Element of dom a
a . f is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a . f) . g is epsilon-transitive epsilon-connected ordinal () set
U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom U is epsilon-transitive epsilon-connected ordinal () set
g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom g is epsilon-transitive epsilon-connected ordinal () set
o is set
U . o is set
g . o is set
{ ((a . b₁) . o) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ in dom a } is set
union { ((a . b₁) . o) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom a : b₁ in dom a } is set
a is epsilon-transitive non empty subset-closed Tarski universal set
On a is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On a),(On a):] is Relation-like non empty set
bool [:(On a),(On a):] is non empty set
U is Relation-like Function-like T-Sequence-like () set
dom U is epsilon-transitive epsilon-connected ordinal () set
(U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
g is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding Element of bool [:(On a),(On a):]
dom g is epsilon-transitive epsilon-connected ordinal non empty () set
rng (U) is () set
o is set
f is set
(U) . f is set
a is epsilon-transitive epsilon-connected ordinal () set
{ ((U . b₁) . a) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ in dom U } is set
union { ((U . b₁) . a) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ in dom U } is set
y is epsilon-transitive epsilon-connected ordinal non empty () Element of a
{ H₁(b₁) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of y : b₁ in y } is set
card { H₁(b₁) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of y : b₁ in y } is epsilon-transitive epsilon-connected ordinal cardinal () set
card y is epsilon-transitive epsilon-connected ordinal non empty cardinal () set
card { ((U . b₁) . a) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom U : b₁ in dom U } is epsilon-transitive epsilon-connected ordinal cardinal () set
card a is epsilon-transitive epsilon-connected ordinal non empty cardinal () set
c is set
x is epsilon-transitive epsilon-connected ordinal () Element of y
U . x is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U . x) . a is epsilon-transitive epsilon-connected ordinal () set
y is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding Element of bool [:(On a),(On a):]
(a,(On a),y,a) is epsilon-transitive epsilon-connected ordinal () Element of a
c is epsilon-transitive epsilon-connected ordinal () set
a is set
the Relation-like Function-like T-Sequence-like Ordinal-yielding set is Relation-like Function-like T-Sequence-like Ordinal-yielding set
the Relation-like Function-like T-Sequence-like () set is Relation-like Function-like T-Sequence-like () set
a is epsilon-transitive non empty subset-closed Tarski universal set
On a is epsilon-transitive epsilon-connected ordinal non empty () set
(a,omega) is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding Element of bool [:(On a),(On a):]
[:(On a),(On a):] is Relation-like non empty set
bool [:(On a),(On a):] is non empty set
U is epsilon-transitive epsilon-connected ordinal non trivial () set
(a,U) is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding Element of bool [:(On a),(On a):]
g is Relation-like Function-like T-Sequence-like set
dom g is epsilon-transitive epsilon-connected ordinal () set
g . {} is set
o is epsilon-transitive epsilon-connected ordinal () set
g . o is set
succ o is epsilon-transitive epsilon-connected ordinal non empty () set
{o} is non empty trivial finite 1 -element set
o \/ {o} is non empty set
g . (succ o) is set
f is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(f) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((f)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom (f) is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (f) : b₁ is_a_fixpoint_of (f) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (f) : b₁ is_a_fixpoint_of (f) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (f) : b₁ is_a_fixpoint_of (f) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (f) : b₁ is_a_fixpoint_of (f) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (f) : b₁ is_a_fixpoint_of (f) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (f) : b₁ is_a_fixpoint_of (f) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (f) : b₁ is_a_fixpoint_of (f) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (f) : b₁ is_a_fixpoint_of (f) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (f) : b₁ is_a_fixpoint_of (f) } ))) is Relation-like Function-like set
(f) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom f is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom f : b₁ is_a_fixpoint_of f } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom f : b₁ is_a_fixpoint_of f } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom f : b₁ is_a_fixpoint_of f } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom f : b₁ is_a_fixpoint_of f } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom f : b₁ is_a_fixpoint_of f } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom f : b₁ is_a_fixpoint_of f } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom f : b₁ is_a_fixpoint_of f } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom f : b₁ is_a_fixpoint_of f } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom f : b₁ is_a_fixpoint_of f } ))) is Relation-like Function-like set
o is epsilon-transitive epsilon-connected ordinal () set
g . o is set
g | o is Relation-like rng g -valued Function-like T-Sequence-like set
rng g is set
((g | o)) is Relation-like Function-like T-Sequence-like () set
(((g | o))) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((g | o)) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (((g | o)) . 0) is epsilon-transitive epsilon-connected ordinal () set
rng ((g | o)) is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g | o)) . 0) : ( b₁ in dom (((g | o)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g | o)) or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g | o)) . 0) : ( b₁ in dom (((g | o)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g | o)) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g | o)) . 0) : ( b₁ in dom (((g | o)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g | o)) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g | o)) . 0) : ( b₁ in dom (((g | o)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g | o)) or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g | o)) . 0) : ( b₁ in dom (((g | o)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g | o)) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g | o)) . 0) : ( b₁ in dom (((g | o)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g | o)) or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g | o)) . 0) : ( b₁ in dom (((g | o)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g | o)) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g | o)) . 0) : ( b₁ in dom (((g | o)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g | o)) or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g | o)) . 0) : ( b₁ in dom (((g | o)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g | o)) or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
o is set
rng g is set
f is set
g . f is set
o is Relation-like Function-like T-Sequence-like () set
dom o is epsilon-transitive epsilon-connected ordinal () set
o . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
0-element_of a is epsilon-transitive epsilon-connected ordinal () Element of a
f is epsilon-transitive epsilon-connected ordinal () set
succ f is epsilon-transitive epsilon-connected ordinal non empty () set
{f} is non empty trivial finite 1 -element set
f \/ {f} is non empty set
o . f is Relation-like Function-like T-Sequence-like Ordinal-yielding set
o . (succ f) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((a)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom (a) is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a) : b₁ is_a_fixpoint_of (a) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a) : b₁ is_a_fixpoint_of (a) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a) : b₁ is_a_fixpoint_of (a) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a) : b₁ is_a_fixpoint_of (a) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a) : b₁ is_a_fixpoint_of (a) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a) : b₁ is_a_fixpoint_of (a) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a) : b₁ is_a_fixpoint_of (a) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a) : b₁ is_a_fixpoint_of (a) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a) : b₁ is_a_fixpoint_of (a) } ))) is Relation-like Function-like set
((o . f)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom (o . f) is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (o . f) : b₁ is_a_fixpoint_of o . f } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (o . f) : b₁ is_a_fixpoint_of o . f } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (o . f) : b₁ is_a_fixpoint_of o . f } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (o . f) : b₁ is_a_fixpoint_of o . f } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (o . f) : b₁ is_a_fixpoint_of o . f } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (o . f) : b₁ is_a_fixpoint_of o . f } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (o . f) : b₁ is_a_fixpoint_of o . f } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (o . f) : b₁ is_a_fixpoint_of o . f } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (o . f) : b₁ is_a_fixpoint_of o . f } ))) is Relation-like Function-like set
f is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite () set
o . f is Relation-like Function-like T-Sequence-like Ordinal-yielding set
o | f is Relation-like rng o -valued Function-like T-Sequence-like () set
rng o is set
((o | f)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(o | f) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o | f) . 0) is epsilon-transitive epsilon-connected ordinal () set
rng (o | f) is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o | f) . 0) : ( b₁ in dom ((o | f) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (o | f) or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o | f) . 0) : ( b₁ in dom ((o | f) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (o | f) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o | f) . 0) : ( b₁ in dom ((o | f) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (o | f) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o | f) . 0) : ( b₁ in dom ((o | f) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (o | f) or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o | f) . 0) : ( b₁ in dom ((o | f) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (o | f) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o | f) . 0) : ( b₁ in dom ((o | f) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (o | f) or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o | f) . 0) : ( b₁ in dom ((o | f) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (o | f) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o | f) . 0) : ( b₁ in dom ((o | f) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (o | f) or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o | f) . 0) : ( b₁ in dom ((o | f) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (o | f) or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
((o | f)) is Relation-like Function-like T-Sequence-like () set
(((o | f))) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((o | f)) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (((o | f)) . 0) is epsilon-transitive epsilon-connected ordinal () set
rng ((o | f)) is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o | f)) . 0) : ( b₁ in dom (((o | f)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o | f)) or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o | f)) . 0) : ( b₁ in dom (((o | f)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o | f)) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o | f)) . 0) : ( b₁ in dom (((o | f)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o | f)) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o | f)) . 0) : ( b₁ in dom (((o | f)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o | f)) or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o | f)) . 0) : ( b₁ in dom (((o | f)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o | f)) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o | f)) . 0) : ( b₁ in dom (((o | f)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o | f)) or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o | f)) . 0) : ( b₁ in dom (((o | f)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o | f)) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o | f)) . 0) : ( b₁ in dom (((o | f)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o | f)) or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o | f)) . 0) : ( b₁ in dom (((o | f)) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o | f)) or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
U is Relation-like Function-like T-Sequence-like () set
dom U is epsilon-transitive epsilon-connected ordinal () set
U . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g is Relation-like Function-like T-Sequence-like () set
dom g is epsilon-transitive epsilon-connected ordinal () set
g . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U . {} is Relation-like Function-like T-Sequence-like Ordinal-yielding set
o is epsilon-transitive epsilon-connected ordinal () set
succ o is epsilon-transitive epsilon-connected ordinal non empty () set
{o} is non empty trivial finite 1 -element set
o \/ {o} is non empty set
U . (succ o) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((U . o)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom (U . o) is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . o) : b₁ is_a_fixpoint_of U . o } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . o) : b₁ is_a_fixpoint_of U . o } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . o) : b₁ is_a_fixpoint_of U . o } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . o) : b₁ is_a_fixpoint_of U . o } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . o) : b₁ is_a_fixpoint_of U . o } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . o) : b₁ is_a_fixpoint_of U . o } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . o) : b₁ is_a_fixpoint_of U . o } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . o) : b₁ is_a_fixpoint_of U . o } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (U . o) : b₁ is_a_fixpoint_of U . o } ))) is Relation-like Function-like set
o is epsilon-transitive epsilon-connected ordinal () set
U . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
U | o is Relation-like rng U -valued Function-like T-Sequence-like () set
rng U is set
((U | o)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U | o) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((U | o) . 0) is epsilon-transitive epsilon-connected ordinal () set
rng (U | o) is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U | o) . 0) : ( b₁ in dom ((U | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (U | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U | o) . 0) : ( b₁ in dom ((U | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (U | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U | o) . 0) : ( b₁ in dom ((U | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (U | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U | o) . 0) : ( b₁ in dom ((U | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (U | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U | o) . 0) : ( b₁ in dom ((U | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (U | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U | o) . 0) : ( b₁ in dom ((U | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (U | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U | o) . 0) : ( b₁ in dom ((U | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (U | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U | o) . 0) : ( b₁ in dom ((U | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (U | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U | o) . 0) : ( b₁ in dom ((U | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (U | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
g . {} is Relation-like Function-like T-Sequence-like Ordinal-yielding set
o is epsilon-transitive epsilon-connected ordinal () set
succ o is epsilon-transitive epsilon-connected ordinal non empty () set
{o} is non empty trivial finite 1 -element set
o \/ {o} is non empty set
g . (succ o) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((g . o)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom (g . o) is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . o) : b₁ is_a_fixpoint_of g . o } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . o) : b₁ is_a_fixpoint_of g . o } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . o) : b₁ is_a_fixpoint_of g . o } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . o) : b₁ is_a_fixpoint_of g . o } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . o) : b₁ is_a_fixpoint_of g . o } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . o) : b₁ is_a_fixpoint_of g . o } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . o) : b₁ is_a_fixpoint_of g . o } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . o) : b₁ is_a_fixpoint_of g . o } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (g . o) : b₁ is_a_fixpoint_of g . o } ))) is Relation-like Function-like set
o is epsilon-transitive epsilon-connected ordinal () set
g . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g | o is Relation-like rng g -valued Function-like T-Sequence-like () set
rng g is set
((g | o)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(g | o) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((g | o) . 0) is epsilon-transitive epsilon-connected ordinal () set
rng (g | o) is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g | o) . 0) : ( b₁ in dom ((g | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (g | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g | o) . 0) : ( b₁ in dom ((g | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (g | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g | o) . 0) : ( b₁ in dom ((g | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (g | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g | o) . 0) : ( b₁ in dom ((g | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (g | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g | o) . 0) : ( b₁ in dom ((g | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (g | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g | o) . 0) : ( b₁ in dom ((g | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (g | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g | o) . 0) : ( b₁ in dom ((g | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (g | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g | o) . 0) : ( b₁ in dom ((g | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (g | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g | o) . 0) : ( b₁ in dom ((g | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng (g | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
Tarski-Class omega is epsilon-transitive non empty subset-closed Tarski universal set
a is epsilon-transitive non empty subset-closed Tarski universal set
card omega is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite cardinal () set
card a is epsilon-transitive epsilon-connected ordinal non empty cardinal () set
a is epsilon-transitive non empty subset-closed Tarski universal set
card a is epsilon-transitive epsilon-connected ordinal non empty cardinal () set
On a is epsilon-transitive epsilon-connected ordinal non empty () set
card omega is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite cardinal () set
card a is epsilon-transitive epsilon-connected ordinal non empty cardinal () set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive non empty subset-closed Tarski universal set
(o) is Relation-like Function-like T-Sequence-like () set
(o) . U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o) . U) is epsilon-transitive epsilon-connected ordinal () set
((o) . U) . g is epsilon-transitive epsilon-connected ordinal () set
(o) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(o) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o) . 0) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
((o) . 0) . b is epsilon-transitive epsilon-connected ordinal () set
(o) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is epsilon-transitive epsilon-connected ordinal () set
(o) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o) . a) is epsilon-transitive epsilon-connected ordinal () set
succ a is epsilon-transitive epsilon-connected ordinal non empty () set
{a} is non empty trivial finite 1 -element set
a \/ {a} is non empty set
(o) . (succ a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o) . (succ a)) is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
(o) . b is Relation-like Function-like T-Sequence-like Ordinal-yielding set
y is epsilon-transitive epsilon-connected ordinal () set
((o) . (succ a)) . y is epsilon-transitive epsilon-connected ordinal () set
On o is epsilon-transitive epsilon-connected ordinal non empty () set
(((o) . a)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } ))) is Relation-like Function-like set
((o) . a) . (((o) . (succ a)) . y) is epsilon-transitive epsilon-connected ordinal () set
dom (o) is epsilon-transitive epsilon-connected ordinal () set
On o is epsilon-transitive epsilon-connected ordinal non empty () set
a is epsilon-transitive epsilon-connected ordinal () set
(o) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o) . a) is epsilon-transitive epsilon-connected ordinal () set
(o) | a is Relation-like rng (o) -valued Function-like T-Sequence-like () set
rng (o) is set
(((o) | a)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((o) | a) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (((o) | a) . 0) is epsilon-transitive epsilon-connected ordinal () set
rng ((o) | a) is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom ((o) | a) is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
(o) . b is Relation-like Function-like T-Sequence-like Ordinal-yielding set
y is epsilon-transitive epsilon-connected ordinal () set
((o) . a) . y is epsilon-transitive epsilon-connected ordinal () set
((o) | a) . b is Relation-like Function-like T-Sequence-like Ordinal-yielding set
y is set
z is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0)
rng ((o) . a) is () set
y is () set
z is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0)
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite () set
g is epsilon-transitive non empty subset-closed Tarski universal set
(g) is Relation-like Function-like T-Sequence-like () set
(g) | U is Relation-like rng (g) -valued Function-like T-Sequence-like () set
rng (g) is set
(((g) | U)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((g) | U) is epsilon-transitive epsilon-connected ordinal () set
{ ((((g) | U) . b₁) . a) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) | U) : b₁ in dom ((g) | U) } is set
union { ((((g) | U) . b₁) . a) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) | U) : b₁ in dom ((g) | U) } is set
(g) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
y is epsilon-transitive epsilon-connected ordinal non trivial () Element of g
(g,y) is Relation-like On g -defined On g -valued Function-like T-Sequence-like non empty V26( On g) V30( On g, On g) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On g),(On g):]
On g is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On g),(On g):] is Relation-like non empty set
bool [:(On g),(On g):] is non empty set
c is Relation-like On g -defined On g -valued Function-like T-Sequence-like non empty V26( On g) V30( On g, On g) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On g),(On g):]
((g) | U) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (((g) | U)) is epsilon-transitive epsilon-connected ordinal () set
dom c is epsilon-transitive epsilon-connected ordinal non empty () set
(g,(On g),c,a) is epsilon-transitive epsilon-connected ordinal () Element of g
dom (g) is epsilon-transitive epsilon-connected ordinal () set
(((g) | U)) . a is epsilon-transitive epsilon-connected ordinal () set
{a} is non empty trivial finite 1 -element set
y is set
z is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) | U)
((g) | U) . z is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((g) | U) . z) . a is epsilon-transitive epsilon-connected ordinal () set
(g) . z is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive non empty subset-closed Tarski universal set
(o) is Relation-like Function-like T-Sequence-like () set
(o) . U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o) . U) is epsilon-transitive epsilon-connected ordinal () set
(o) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((o) . a) . g is epsilon-transitive epsilon-connected ordinal () set
((o) . U) . g is epsilon-transitive epsilon-connected ordinal () set
(o) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o) . 0) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
(o) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((o) . a) . b is epsilon-transitive epsilon-connected ordinal () set
((o) . 0) . b is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
(o) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o) . a) is epsilon-transitive epsilon-connected ordinal () set
succ a is epsilon-transitive epsilon-connected ordinal non empty () set
{a} is non empty trivial finite 1 -element set
a \/ {a} is non empty set
(o) . (succ a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o) . (succ a)) is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
(o) . b is Relation-like Function-like T-Sequence-like Ordinal-yielding set
y is epsilon-transitive epsilon-connected ordinal () set
((o) . b) . y is epsilon-transitive epsilon-connected ordinal () set
((o) . (succ a)) . y is epsilon-transitive epsilon-connected ordinal () set
On o is epsilon-transitive epsilon-connected ordinal non empty () set
(((o) . a)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) . a) : b₁ is_a_fixpoint_of (o) . a } ))) is Relation-like Function-like set
((o) . a) . y is epsilon-transitive epsilon-connected ordinal () set
c is epsilon-transitive epsilon-connected ordinal () set
(o) . c is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is epsilon-transitive epsilon-connected ordinal () set
(o) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o) . a) is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
(o) . b is Relation-like Function-like T-Sequence-like Ordinal-yielding set
y is epsilon-transitive epsilon-connected ordinal () set
((o) . b) . y is epsilon-transitive epsilon-connected ordinal () set
((o) . a) . y is epsilon-transitive epsilon-connected ordinal () set
On o is epsilon-transitive epsilon-connected ordinal non empty () set
(o) | a is Relation-like rng (o) -valued Function-like T-Sequence-like () set
rng (o) is set
(((o) | a)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((o) | a) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (((o) | a) . 0) is epsilon-transitive epsilon-connected ordinal () set
rng ((o) | a) is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((o) | a) . 0) : ( b₁ in dom (((o) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((o) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom (o) is epsilon-transitive epsilon-connected ordinal () set
dom ((o) | a) is epsilon-transitive epsilon-connected ordinal () set
c is epsilon-transitive epsilon-connected ordinal () set
((o) | a) . c is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(o) . c is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((o) | a) . b is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite () set
o is epsilon-transitive non empty subset-closed Tarski universal set
(o) is Relation-like Function-like T-Sequence-like () set
On o is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On o),(On o):] is Relation-like non empty set
bool [:(On o),(On o):] is non empty set
(o) | g is Relation-like rng (o) -valued Function-like T-Sequence-like () set
rng (o) is set
(((o) | g)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((o) | g)) . a is epsilon-transitive epsilon-connected ordinal () set
(o) . U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((o) | g) is epsilon-transitive epsilon-connected ordinal () set
{ ((((o) | g) . b₁) . a) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) | g) : b₁ in dom ((o) | g) } is set
union { ((((o) | g) . b₁) . a) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) | g) : b₁ in dom ((o) | g) } is set
(o) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
c is Relation-like On o -defined On o -valued Function-like T-Sequence-like non empty V26( On o) V30( On o, On o) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On o),(On o):]
((o) | g) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
x is Relation-like On o -defined On o -valued Function-like T-Sequence-like non empty V26( On o) V30( On o, On o) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On o),(On o):]
((o) | g) . U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (((o) | g)) is epsilon-transitive epsilon-connected ordinal () set
dom c is epsilon-transitive epsilon-connected ordinal non empty () set
dom (o) is epsilon-transitive epsilon-connected ordinal () set
(((o) | g) . U) . a is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
((o) | g) . y is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(o) . y is Relation-like Function-like T-Sequence-like Ordinal-yielding set
y is Relation-like On o -defined On o -valued Function-like T-Sequence-like non empty V26( On o) V30( On o, On o) Ordinal-yielding Element of bool [:(On o),(On o):]
(o,(On o),y,a) is epsilon-transitive epsilon-connected ordinal () Element of o
dom x is epsilon-transitive epsilon-connected ordinal non empty () set
z is set
h is epsilon-transitive epsilon-connected ordinal () Element of dom ((o) | g)
((o) | g) . h is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((o) | g) . h) . a is epsilon-transitive epsilon-connected ordinal () set
(o) . h is Relation-like Function-like T-Sequence-like Ordinal-yielding set
f2 is Relation-like On o -defined On o -valued Function-like T-Sequence-like non empty V26( On o) V30( On o, On o) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On o),(On o):]
dom f2 is epsilon-transitive epsilon-connected ordinal non empty () set
y is epsilon-transitive epsilon-connected ordinal () set
(o) . y is Relation-like Function-like T-Sequence-like Ordinal-yielding set
succ U is epsilon-transitive epsilon-connected ordinal non empty () set
{U} is non empty trivial finite 1 -element set
U \/ {U} is non empty set
((o) | g) . (succ U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((o) | g) . (succ U)) . a is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
(o) . (succ U) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
f1 is Relation-like On o -defined On o -valued Function-like T-Sequence-like non empty V26( On o) V30( On o, On o) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On o),(On o):]
(x) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing continuous non-decreasing () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom x : b₁ is_a_fixpoint_of x } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom x : b₁ is_a_fixpoint_of x } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom x : b₁ is_a_fixpoint_of x } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom x : b₁ is_a_fixpoint_of x } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom x : b₁ is_a_fixpoint_of x } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom x : b₁ is_a_fixpoint_of x } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom x : b₁ is_a_fixpoint_of x } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom x : b₁ is_a_fixpoint_of x } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom x : b₁ is_a_fixpoint_of x } ))) is Relation-like Function-like set
dom f1 is epsilon-transitive epsilon-connected ordinal non empty () set
(o,(On o),x,a) is epsilon-transitive epsilon-connected ordinal () Element of o
y is set
dom ((o) . U) is epsilon-transitive epsilon-connected ordinal () set
((o) . U) . ((((o) | g)) . a) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive non empty subset-closed Tarski universal set
(a) is Relation-like Function-like T-Sequence-like () set
(a) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
On a is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On a),(On a):] is Relation-like non empty set
bool [:(On a),(On a):] is non empty set
U is epsilon-transitive epsilon-connected ordinal non trivial () Element of a
(a,U) is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On a),(On a):]
a is epsilon-transitive epsilon-connected ordinal () set
succ a is epsilon-transitive epsilon-connected ordinal non empty () set
{a} is non empty trivial finite 1 -element set
a \/ {a} is non empty set
U is epsilon-transitive non empty subset-closed Tarski universal set
(U) is Relation-like Function-like T-Sequence-like () set
(U) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
On U is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On U),(On U):] is Relation-like non empty set
bool [:(On U),(On U):] is non empty set
(U) . (succ a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g is Relation-like On U -defined On U -valued Function-like T-Sequence-like non empty V26( On U) V30( On U, On U) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On U),(On U):]
(g) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing continuous non-decreasing () set
dom g is epsilon-transitive epsilon-connected ordinal non empty () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ))) is Relation-like Function-like set
a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite () set
U is epsilon-transitive non empty subset-closed Tarski universal set
(U) is Relation-like Function-like T-Sequence-like () set
On U is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On U),(On U):] is Relation-like non empty set
bool [:(On U),(On U):] is non empty set
(U) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) | a is Relation-like rng (U) -valued Function-like T-Sequence-like () set
rng (U) is set
(((U) | a)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((U) | a) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (((U) | a) . 0) is epsilon-transitive epsilon-connected ordinal () set
rng ((U) | a) is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
dom ((U) | a) is epsilon-transitive epsilon-connected ordinal () set
dom ((U) . a) is epsilon-transitive epsilon-connected ordinal () set
(U) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
f is Relation-like On U -defined On U -valued Function-like T-Sequence-like non empty V26( On U) V30( On U, On U) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On U),(On U):]
dom f is epsilon-transitive epsilon-connected ordinal non empty () set
a is epsilon-transitive epsilon-connected ordinal () set
((U) | a) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((U) | a)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is Relation-like On U -defined On U -valued Function-like T-Sequence-like non empty V26( On U) V30( On U, On U) Ordinal-yielding Element of bool [:(On U),(On U):]
dom a is epsilon-transitive epsilon-connected ordinal non empty () set
rng a is non empty () set
rng ((U) . a) is () set
b is set
y is set
(U,(On U),a,y) is epsilon-transitive epsilon-connected ordinal () Element of U
c is epsilon-transitive epsilon-connected ordinal () set
y is Relation-like Function-like T-Sequence-like Ordinal-yielding set
z is set
((U) | a) . z is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) . z is Relation-like Function-like T-Sequence-like Ordinal-yielding set
Union a is epsilon-transitive epsilon-connected ordinal () set
union (rng a) is epsilon-transitive epsilon-connected ordinal () set
Union ((U) . a) is epsilon-transitive epsilon-connected ordinal () set
union (rng ((U) . a)) is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
succ b is epsilon-transitive epsilon-connected ordinal non empty () set
{b} is non empty trivial finite 1 -element set
b \/ {b} is non empty set
{ ((((U) | a) . b₁) . (succ b)) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) | a) : b₁ in dom ((U) | a) } is set
(U,(On U),a,(succ b)) is epsilon-transitive epsilon-connected ordinal () Element of U
union { ((((U) | a) . b₁) . (succ b)) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) | a) : b₁ in dom ((U) | a) } is set
(U,(On U),f,(succ b)) is epsilon-transitive epsilon-connected ordinal () Element of U
(U,(On U),f,b) is epsilon-transitive epsilon-connected ordinal () Element of U
b is set
y is set
((U) . a) . y is set
(U,(On U),f,b) is epsilon-transitive epsilon-connected ordinal () Element of U
b is epsilon-transitive epsilon-connected ordinal () Element of U
card b is epsilon-transitive epsilon-connected ordinal cardinal () set
card U is epsilon-transitive epsilon-connected ordinal non empty cardinal () set
card (rng ((U) . a)) is epsilon-transitive epsilon-connected ordinal cardinal () set
y is Element of U
union y is Element of U
rng ((U) . a) is () set
g is set
o is set
((U) . a) . o is set
(U) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom ((U) . 0) is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
((U) | a) . g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) . g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive non empty subset-closed Tarski universal set
(U) is Relation-like Function-like T-Sequence-like () set
(U) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
On U is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On U),(On U):] is Relation-like non empty set
bool [:(On U),(On U):] is non empty set
(U) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g is epsilon-transitive epsilon-connected ordinal () set
(U) . g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
succ g is epsilon-transitive epsilon-connected ordinal non empty () set
{g} is non empty trivial finite 1 -element set
g \/ {g} is non empty set
(U) . (succ g) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g is epsilon-transitive epsilon-connected ordinal () set
(U) . g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
o is epsilon-transitive epsilon-connected ordinal () set
(U) . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive non empty subset-closed Tarski universal set
(U) is Relation-like Function-like T-Sequence-like () set
(U) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
g is epsilon-transitive non empty subset-closed Tarski universal set
(g) is Relation-like Function-like T-Sequence-like () set
(g) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
On U is epsilon-transitive epsilon-connected ordinal non empty () set
On g is epsilon-transitive epsilon-connected ordinal non empty () set
(U) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U,omega) is Relation-like On U -defined On U -valued Function-like T-Sequence-like non empty V26( On U) V30( On U, On U) Ordinal-yielding Element of bool [:(On U),(On U):]
[:(On U),(On U):] is Relation-like non empty set
bool [:(On U),(On U):] is non empty set
(g) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(g,omega) is Relation-like On g -defined On g -valued Function-like T-Sequence-like non empty V26( On g) V30( On g, On g) Ordinal-yielding Element of bool [:(On g),(On g):]
[:(On g),(On g):] is Relation-like non empty set
bool [:(On g),(On g):] is non empty set
dom (U,omega) is epsilon-transitive epsilon-connected ordinal non empty () set
dom (g,omega) is epsilon-transitive epsilon-connected ordinal non empty () set
o is set
f is epsilon-transitive epsilon-connected ordinal () Element of U
(U,(On U),(U,omega),f) is epsilon-transitive epsilon-connected ordinal () Element of U
a is epsilon-transitive epsilon-connected ordinal () Element of g
exp (omega,a) is epsilon-transitive epsilon-connected ordinal () set
(U,(On U),(U,omega),o) is epsilon-transitive epsilon-connected ordinal () Element of U
(g,(On g),(g,omega),o) is epsilon-transitive epsilon-connected ordinal () Element of g
o is epsilon-transitive epsilon-connected ordinal () set
(U) . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(g) . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
succ o is epsilon-transitive epsilon-connected ordinal non empty () set
{o} is non empty trivial finite 1 -element set
o \/ {o} is non empty set
(U) . (succ o) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(g) . (succ o) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((U) . o)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom ((U) . o) is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) . o) : b₁ is_a_fixpoint_of (U) . o } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) . o) : b₁ is_a_fixpoint_of (U) . o } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) . o) : b₁ is_a_fixpoint_of (U) . o } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) . o) : b₁ is_a_fixpoint_of (U) . o } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) . o) : b₁ is_a_fixpoint_of (U) . o } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) . o) : b₁ is_a_fixpoint_of (U) . o } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) . o) : b₁ is_a_fixpoint_of (U) . o } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) . o) : b₁ is_a_fixpoint_of (U) . o } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) . o) : b₁ is_a_fixpoint_of (U) . o } ))) is Relation-like Function-like set
(((g) . o)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom ((g) . o) is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) . o) : b₁ is_a_fixpoint_of (g) . o } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) . o) : b₁ is_a_fixpoint_of (g) . o } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) . o) : b₁ is_a_fixpoint_of (g) . o } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) . o) : b₁ is_a_fixpoint_of (g) . o } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) . o) : b₁ is_a_fixpoint_of (g) . o } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) . o) : b₁ is_a_fixpoint_of (g) . o } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) . o) : b₁ is_a_fixpoint_of (g) . o } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) . o) : b₁ is_a_fixpoint_of (g) . o } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((g) . o) : b₁ is_a_fixpoint_of (g) . o } ))) is Relation-like Function-like set
o is epsilon-transitive epsilon-connected ordinal () set
(U) . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(g) . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) | o is Relation-like rng (U) -valued Function-like T-Sequence-like () set
rng (U) is set
(g) | o is Relation-like rng (g) -valued Function-like T-Sequence-like () set
rng (g) is set
(((U) | o)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((U) | o) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (((U) | o) . 0) is epsilon-transitive epsilon-connected ordinal () set
rng ((U) | o) is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | o) . 0) : ( b₁ in dom (((U) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | o) . 0) : ( b₁ in dom (((U) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | o) . 0) : ( b₁ in dom (((U) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | o) . 0) : ( b₁ in dom (((U) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | o) . 0) : ( b₁ in dom (((U) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | o) . 0) : ( b₁ in dom (((U) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | o) . 0) : ( b₁ in dom (((U) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | o) . 0) : ( b₁ in dom (((U) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | o) . 0) : ( b₁ in dom (((U) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
(((g) | o)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((g) | o) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (((g) | o) . 0) is epsilon-transitive epsilon-connected ordinal () set
rng ((g) | o) is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g) | o) . 0) : ( b₁ in dom (((g) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g) | o) . 0) : ( b₁ in dom (((g) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g) | o) . 0) : ( b₁ in dom (((g) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g) | o) . 0) : ( b₁ in dom (((g) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g) | o) . 0) : ( b₁ in dom (((g) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g) | o) . 0) : ( b₁ in dom (((g) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g) | o) . 0) : ( b₁ in dom (((g) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g) | o) . 0) : ( b₁ in dom (((g) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((g) | o) . 0) : ( b₁ in dom (((g) | o) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((g) | o) or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
dom (g) is epsilon-transitive epsilon-connected ordinal () set
dom ((U) | o) is epsilon-transitive epsilon-connected ordinal () set
dom ((g) | o) is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
((U) | o) . b is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) . b is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((g) | o) . b is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(g) . b is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive non empty subset-closed Tarski universal set
(g) is Relation-like Function-like T-Sequence-like () set
(g) . U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((g) . U) . a is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive non empty subset-closed Tarski universal set
(o) is Relation-like Function-like T-Sequence-like () set
(o) . U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((o) . U) . a is epsilon-transitive epsilon-connected ordinal () set
On g is epsilon-transitive epsilon-connected ordinal non empty () set
On o is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On o),(On o):] is Relation-like non empty set
bool [:(On o),(On o):] is non empty set
[:(On g),(On g):] is Relation-like non empty set
bool [:(On g),(On g):] is non empty set
dom ((g) . U) is epsilon-transitive epsilon-connected ordinal () set
dom ((o) . U) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite () set
U is epsilon-transitive non empty subset-closed Tarski universal set
(U) is Relation-like Function-like T-Sequence-like () set
On U is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On U),(On U):] is Relation-like non empty set
bool [:(On U),(On U):] is non empty set
(U) | a is Relation-like rng (U) -valued Function-like T-Sequence-like () set
rng (U) is set
(((U) | a)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((U) | a)) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((U) | a) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
dom (((U) | a) . 0) is epsilon-transitive epsilon-connected ordinal () set
rng ((U) | a) is set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((U) | a) . 0) : ( b₁ in dom (((U) | a) . 0) & ( for b₂ being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
( not b₂ in rng ((U) | a) or b₁ is_a_fixpoint_of b₂ ) ) ) } ))) is Relation-like Function-like set
dom (U) is epsilon-transitive epsilon-connected ordinal () set
dom ((U) | a) is epsilon-transitive epsilon-connected ordinal () set
f is epsilon-transitive epsilon-connected ordinal () set
((U) | a) . f is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) . f is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(U) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
f is Relation-like On U -defined On U -valued Function-like T-Sequence-like non empty V26( On U) V30( On U, On U) Ordinal-yielding Element of bool [:(On U),(On U):]
dom f is epsilon-transitive epsilon-connected ordinal non empty () set
{ ((((U) | a) . b₁) . a₁) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) | a) : b₁ in dom ((U) | a) } is set
b is epsilon-transitive epsilon-connected ordinal () set
f . b is epsilon-transitive epsilon-connected ordinal () set
y is epsilon-transitive epsilon-connected ordinal () set
f . y is epsilon-transitive epsilon-connected ordinal () set
(U,(On U),f,b) is epsilon-transitive epsilon-connected ordinal () Element of U
{ ((((U) | a) . b₁) . b) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) | a) : b₁ in dom ((U) | a) } is set
union H₁(b) is set
(U,(On U),f,y) is epsilon-transitive epsilon-connected ordinal () Element of U
{ ((((U) | a) . b₁) . y) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) | a) : b₁ in dom ((U) | a) } is set
union H₁(y) is set
c is epsilon-transitive epsilon-connected ordinal () set
x is set
y is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) | a)
((U) | a) . y is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((U) | a) . y) . b is epsilon-transitive epsilon-connected ordinal () set
(U) . y is Relation-like Function-like T-Sequence-like Ordinal-yielding set
z is Relation-like On U -defined On U -valued Function-like T-Sequence-like non empty V26( On U) V30( On U, On U) Ordinal-yielding increasing non-decreasing Element of bool [:(On U),(On U):]
dom z is epsilon-transitive epsilon-connected ordinal non empty () set
(U,(On U),z,b) is epsilon-transitive epsilon-connected ordinal () Element of U
(U,(On U),z,y) is epsilon-transitive epsilon-connected ordinal () Element of U
b is epsilon-transitive epsilon-connected ordinal () set
f . b is epsilon-transitive epsilon-connected ordinal () set
f | b is Relation-like b -defined On U -defined On U -valued rng f -valued Function-like T-Sequence-like Ordinal-yielding set
rng f is non empty () set
y is epsilon-transitive epsilon-connected ordinal () set
(U,(On U),f,b) is epsilon-transitive epsilon-connected ordinal () Element of U
{ ((((U) | a) . b₁) . b) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) | a) : b₁ in dom ((U) | a) } is set
union H₁(b) is set
f | b is Relation-like On U -defined b -defined On U -defined On U -valued rng f -valued Function-like T-Sequence-like Ordinal-yielding Element of bool [:(On U),(On U):]
dom (f | b) is epsilon-transitive epsilon-connected ordinal () set
Union (f | b) is epsilon-transitive epsilon-connected ordinal () set
rng (f | b) is () set
union (rng (f | b)) is epsilon-transitive epsilon-connected ordinal () set
c is epsilon-transitive epsilon-connected ordinal () set
x is set
y is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) | a)
((U) | a) . y is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((U) | a) . y) . b is epsilon-transitive epsilon-connected ordinal () set
(U) . y is Relation-like Function-like T-Sequence-like Ordinal-yielding set
z is Relation-like On U -defined On U -valued Function-like T-Sequence-like non empty V26( On U) V30( On U, On U) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On U),(On U):]
dom z is epsilon-transitive epsilon-connected ordinal non empty () set
(U,(On U),z,b) is epsilon-transitive epsilon-connected ordinal () Element of U
z | b is Relation-like On U -defined b -defined On U -defined On U -valued rng z -valued Function-like T-Sequence-like Ordinal-yielding Element of bool [:(On U),(On U):]
rng z is non empty () set
dom (z | b) is epsilon-transitive epsilon-connected ordinal () set
Union (z | b) is epsilon-transitive epsilon-connected ordinal () set
rng (z | b) is () set
union (rng (z | b)) is epsilon-transitive epsilon-connected ordinal () set
lim (z | b) is epsilon-transitive epsilon-connected ordinal () set
h is set
(z | b) . h is set
(U,(On U),z,h) is epsilon-transitive epsilon-connected ordinal () Element of U
{ ((((U) | a) . b₁) . h) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) | a) : b₁ in dom ((U) | a) } is set
union H₁(h) is set
(U,(On U),f,h) is epsilon-transitive epsilon-connected ordinal () Element of U
(f | b) . h is set
c is epsilon-transitive epsilon-connected ordinal () set
x is set
(f | b) . x is set
(U,(On U),f,x) is epsilon-transitive epsilon-connected ordinal () Element of U
{ ((((U) | a) . b₁) . x) where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) | a) : b₁ in dom ((U) | a) } is set
union H₁(x) is set
y is set
z is epsilon-transitive epsilon-connected ordinal () Element of dom ((U) | a)
((U) | a) . z is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((U) | a) . z) . x is set
(U) . z is Relation-like Function-like T-Sequence-like Ordinal-yielding set
h is Relation-like On U -defined On U -valued Function-like T-Sequence-like non empty V26( On U) V30( On U, On U) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On U),(On U):]
dom h is epsilon-transitive epsilon-connected ordinal non empty () set
(U,(On U),h,x) is epsilon-transitive epsilon-connected ordinal () Element of U
(U,(On U),h,b) is epsilon-transitive epsilon-connected ordinal () Element of U
a is epsilon-transitive epsilon-connected ordinal () set
a \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (a \/ omega) is epsilon-transitive non empty subset-closed Tarski universal set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
a \/ U is epsilon-transitive epsilon-connected ordinal () set
(a \/ U) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((a \/ U) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((a \/ U) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((a \/ U) \/ omega))) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((a \/ U) \/ omega))) . a) . U is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal natural finite cardinal () set
U is epsilon-transitive epsilon-connected ordinal () set
(a,U) is epsilon-transitive epsilon-connected ordinal () set
a \/ U is epsilon-transitive epsilon-connected ordinal () set
(a \/ U) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((a \/ U) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((a \/ U) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((a \/ U) \/ omega))) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((a \/ U) \/ omega))) . a) . U is epsilon-transitive epsilon-connected ordinal () set
U \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (U \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (U \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (U \/ omega))) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (U \/ omega))) . a) . U is epsilon-transitive epsilon-connected ordinal () set
a \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
g is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
a \/ U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
(a \/ U) \/ g is epsilon-transitive epsilon-connected ordinal () set
Tarski-Class ((a \/ U) \/ g) is epsilon-transitive non empty subset-closed Tarski universal set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
(U,a) is epsilon-transitive epsilon-connected ordinal () set
U \/ a is epsilon-transitive epsilon-connected ordinal () set
(U \/ a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((U \/ a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((U \/ a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((U \/ a) \/ omega))) . U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((U \/ a) \/ omega))) . U) . a is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive non empty subset-closed Tarski universal set
(g) is Relation-like Function-like T-Sequence-like () set
(g) . U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((g) . U) . a is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
(0,a) is epsilon-transitive epsilon-connected ordinal () set
0 \/ a is epsilon-transitive epsilon-connected ordinal () set
(0 \/ a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((0 \/ a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((0 \/ a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((0 \/ a) \/ omega))) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((0 \/ a) \/ omega))) . 0) . a is epsilon-transitive epsilon-connected ordinal () set
a \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (a \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (a \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (a \/ omega))) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (a \/ omega))) . 0) . a is epsilon-transitive epsilon-connected ordinal () set
exp (omega,a) is epsilon-transitive epsilon-connected ordinal () set
On (Tarski-Class ((0 \/ a) \/ omega)) is epsilon-transitive epsilon-connected ordinal non empty () set
((Tarski-Class ((0 \/ a) \/ omega)),omega) is Relation-like On (Tarski-Class ((0 \/ a) \/ omega)) -defined On (Tarski-Class ((0 \/ a) \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class ((0 \/ a) \/ omega))) V30( On (Tarski-Class ((0 \/ a) \/ omega)), On (Tarski-Class ((0 \/ a) \/ omega))) Ordinal-yielding Element of bool [:(On (Tarski-Class ((0 \/ a) \/ omega))),(On (Tarski-Class ((0 \/ a) \/ omega))):]
[:(On (Tarski-Class ((0 \/ a) \/ omega))),(On (Tarski-Class ((0 \/ a) \/ omega))):] is Relation-like non empty set
bool [:(On (Tarski-Class ((0 \/ a) \/ omega))),(On (Tarski-Class ((0 \/ a) \/ omega))):] is non empty set
((Tarski-Class ((0 \/ a) \/ omega)),(On (Tarski-Class ((0 \/ a) \/ omega))),((Tarski-Class ((0 \/ a) \/ omega)),omega),a) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class ((0 \/ a) \/ omega)
a is epsilon-transitive epsilon-connected ordinal () set
succ a is epsilon-transitive epsilon-connected ordinal non empty () set
{a} is non empty trivial finite 1 -element set
a \/ {a} is non empty set
U is epsilon-transitive epsilon-connected ordinal () set
((succ a),U) is epsilon-transitive epsilon-connected ordinal () set
(succ a) \/ U is epsilon-transitive epsilon-connected ordinal non empty () set
((succ a) \/ U) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (((succ a) \/ U) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (((succ a) \/ U) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (((succ a) \/ U) \/ omega))) . (succ a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (((succ a) \/ U) \/ omega))) . (succ a)) . U is epsilon-transitive epsilon-connected ordinal () set
(a,((succ a),U)) is epsilon-transitive epsilon-connected ordinal () set
a \/ ((succ a),U) is epsilon-transitive epsilon-connected ordinal () set
(a \/ ((succ a),U)) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((a \/ ((succ a),U)) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((a \/ ((succ a),U)) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((a \/ ((succ a),U)) \/ omega))) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((a \/ ((succ a),U)) \/ omega))) . a) . ((succ a),U) is epsilon-transitive epsilon-connected ordinal () set
a \/ U is epsilon-transitive epsilon-connected ordinal () set
(a \/ U) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((a \/ U) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
o is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class ((a \/ U) \/ omega)
succ o is epsilon-transitive epsilon-connected ordinal non empty () Element of Tarski-Class ((a \/ U) \/ omega)
{o} is non empty trivial finite 1 -element set
o \/ {o} is non empty set
On (Tarski-Class ((a \/ U) \/ omega)) is epsilon-transitive epsilon-connected ordinal non empty () set
((Tarski-Class ((a \/ U) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((a \/ U) \/ omega))) . (succ a) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((Tarski-Class ((a \/ U) \/ omega))) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((((Tarski-Class ((a \/ U) \/ omega))) . a)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom (((Tarski-Class ((a \/ U) \/ omega))) . a) is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((Tarski-Class ((a \/ U) \/ omega))) . a) : b₁ is_a_fixpoint_of ((Tarski-Class ((a \/ U) \/ omega))) . a } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((Tarski-Class ((a \/ U) \/ omega))) . a) : b₁ is_a_fixpoint_of ((Tarski-Class ((a \/ U) \/ omega))) . a } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((Tarski-Class ((a \/ U) \/ omega))) . a) : b₁ is_a_fixpoint_of ((Tarski-Class ((a \/ U) \/ omega))) . a } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((Tarski-Class ((a \/ U) \/ omega))) . a) : b₁ is_a_fixpoint_of ((Tarski-Class ((a \/ U) \/ omega))) . a } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((Tarski-Class ((a \/ U) \/ omega))) . a) : b₁ is_a_fixpoint_of ((Tarski-Class ((a \/ U) \/ omega))) . a } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((Tarski-Class ((a \/ U) \/ omega))) . a) : b₁ is_a_fixpoint_of ((Tarski-Class ((a \/ U) \/ omega))) . a } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((Tarski-Class ((a \/ U) \/ omega))) . a) : b₁ is_a_fixpoint_of ((Tarski-Class ((a \/ U) \/ omega))) . a } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((Tarski-Class ((a \/ U) \/ omega))) . a) : b₁ is_a_fixpoint_of ((Tarski-Class ((a \/ U) \/ omega))) . a } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (((Tarski-Class ((a \/ U) \/ omega))) . a) : b₁ is_a_fixpoint_of ((Tarski-Class ((a \/ U) \/ omega))) . a } ))) is Relation-like Function-like set
[:(On (Tarski-Class ((a \/ U) \/ omega))),(On (Tarski-Class ((a \/ U) \/ omega))):] is Relation-like non empty set
bool [:(On (Tarski-Class ((a \/ U) \/ omega))),(On (Tarski-Class ((a \/ U) \/ omega))):] is non empty set
((Tarski-Class ((a \/ U) \/ omega))) . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((Tarski-Class ((a \/ U) \/ omega))) . (succ o) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
f is Relation-like On (Tarski-Class ((a \/ U) \/ omega)) -defined On (Tarski-Class ((a \/ U) \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class ((a \/ U) \/ omega))) V30( On (Tarski-Class ((a \/ U) \/ omega)), On (Tarski-Class ((a \/ U) \/ omega))) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On (Tarski-Class ((a \/ U) \/ omega))),(On (Tarski-Class ((a \/ U) \/ omega))):]
dom f is epsilon-transitive epsilon-connected ordinal non empty () set
a is Relation-like On (Tarski-Class ((a \/ U) \/ omega)) -defined On (Tarski-Class ((a \/ U) \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class ((a \/ U) \/ omega))) V30( On (Tarski-Class ((a \/ U) \/ omega)), On (Tarski-Class ((a \/ U) \/ omega))) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On (Tarski-Class ((a \/ U) \/ omega))),(On (Tarski-Class ((a \/ U) \/ omega))):]
dom a is epsilon-transitive epsilon-connected ordinal non empty () set
((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class ((a \/ U) \/ omega)
a \/ ((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U) is epsilon-transitive epsilon-connected ordinal () set
(a \/ ((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U)) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((a \/ ((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U)) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((succ o),U) is epsilon-transitive epsilon-connected ordinal () set
(succ o) \/ U is epsilon-transitive epsilon-connected ordinal non empty () set
((succ o) \/ U) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (((succ o) \/ U) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (((succ o) \/ U) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (((succ o) \/ U) \/ omega))) . (succ o) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (((succ o) \/ U) \/ omega))) . (succ o)) . U is epsilon-transitive epsilon-connected ordinal () set
(o,((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U)) is epsilon-transitive epsilon-connected ordinal () set
o \/ ((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U) is epsilon-transitive epsilon-connected ordinal () set
(o \/ ((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U)) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((o \/ ((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U)) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((o \/ ((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U)) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((o \/ ((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U)) \/ omega))) . o is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((o \/ ((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U)) \/ omega))) . o) . ((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U) is epsilon-transitive epsilon-connected ordinal () set
((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),f,((Tarski-Class ((a \/ U) \/ omega)),(On (Tarski-Class ((a \/ U) \/ omega))),a,U)) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class ((a \/ U) \/ omega)
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
(U,g) is epsilon-transitive epsilon-connected ordinal () set
U \/ g is epsilon-transitive epsilon-connected ordinal () set
(U \/ g) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((U \/ g) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((U \/ g) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((U \/ g) \/ omega))) . U is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((U \/ g) \/ omega))) . U) . g is epsilon-transitive epsilon-connected ordinal () set
(a,(U,g)) is epsilon-transitive epsilon-connected ordinal () set
a \/ (U,g) is epsilon-transitive epsilon-connected ordinal () set
(a \/ (U,g)) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((a \/ (U,g)) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((a \/ (U,g)) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((a \/ (U,g)) \/ omega))) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((a \/ (U,g)) \/ omega))) . a) . (U,g) is epsilon-transitive epsilon-connected ordinal () set
On (Tarski-Class ((U \/ g) \/ omega)) is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On (Tarski-Class ((U \/ g) \/ omega))),(On (Tarski-Class ((U \/ g) \/ omega))):] is Relation-like non empty set
bool [:(On (Tarski-Class ((U \/ g) \/ omega))),(On (Tarski-Class ((U \/ g) \/ omega))):] is non empty set
((Tarski-Class ((U \/ g) \/ omega))) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
a is Relation-like On (Tarski-Class ((U \/ g) \/ omega)) -defined On (Tarski-Class ((U \/ g) \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class ((U \/ g) \/ omega))) V30( On (Tarski-Class ((U \/ g) \/ omega)), On (Tarski-Class ((U \/ g) \/ omega))) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On (Tarski-Class ((U \/ g) \/ omega))),(On (Tarski-Class ((U \/ g) \/ omega))):]
dom a is epsilon-transitive epsilon-connected ordinal non empty () set
((Tarski-Class ((U \/ g) \/ omega)),(On (Tarski-Class ((U \/ g) \/ omega))),a,g) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class ((U \/ g) \/ omega)
f is Relation-like On (Tarski-Class ((U \/ g) \/ omega)) -defined On (Tarski-Class ((U \/ g) \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class ((U \/ g) \/ omega))) V30( On (Tarski-Class ((U \/ g) \/ omega)), On (Tarski-Class ((U \/ g) \/ omega))) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On (Tarski-Class ((U \/ g) \/ omega))),(On (Tarski-Class ((U \/ g) \/ omega))):]
((Tarski-Class ((U \/ g) \/ omega)),(On (Tarski-Class ((U \/ g) \/ omega))),f,((Tarski-Class ((U \/ g) \/ omega)),(On (Tarski-Class ((U \/ g) \/ omega))),a,g)) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class ((U \/ g) \/ omega)
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
(g,a) is epsilon-transitive epsilon-connected ordinal () set
g \/ a is epsilon-transitive epsilon-connected ordinal () set
(g \/ a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((g \/ a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((g \/ a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((g \/ a) \/ omega))) . g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((g \/ a) \/ omega))) . g) . a is epsilon-transitive epsilon-connected ordinal () set
(g,U) is epsilon-transitive epsilon-connected ordinal () set
g \/ U is epsilon-transitive epsilon-connected ordinal () set
(g \/ U) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((g \/ U) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((g \/ U) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((g \/ U) \/ omega))) . g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((g \/ U) \/ omega))) . g) . U is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal natural finite cardinal () set
On (Tarski-Class ((g \/ U) \/ omega)) is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On (Tarski-Class ((g \/ U) \/ omega))),(On (Tarski-Class ((g \/ U) \/ omega))):] is Relation-like non empty set
bool [:(On (Tarski-Class ((g \/ U) \/ omega))),(On (Tarski-Class ((g \/ U) \/ omega))):] is non empty set
On (Tarski-Class ((g \/ a) \/ omega)) is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On (Tarski-Class ((g \/ a) \/ omega))),(On (Tarski-Class ((g \/ a) \/ omega))):] is Relation-like non empty set
bool [:(On (Tarski-Class ((g \/ a) \/ omega))),(On (Tarski-Class ((g \/ a) \/ omega))):] is non empty set
a is Relation-like On (Tarski-Class ((g \/ U) \/ omega)) -defined On (Tarski-Class ((g \/ U) \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class ((g \/ U) \/ omega))) V30( On (Tarski-Class ((g \/ U) \/ omega)), On (Tarski-Class ((g \/ U) \/ omega))) Ordinal-yielding increasing non-decreasing Element of bool [:(On (Tarski-Class ((g \/ U) \/ omega))),(On (Tarski-Class ((g \/ U) \/ omega))):]
dom a is epsilon-transitive epsilon-connected ordinal non empty () set
b is Relation-like On (Tarski-Class ((g \/ a) \/ omega)) -defined On (Tarski-Class ((g \/ a) \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class ((g \/ a) \/ omega))) V30( On (Tarski-Class ((g \/ a) \/ omega)), On (Tarski-Class ((g \/ a) \/ omega))) Ordinal-yielding increasing non-decreasing Element of bool [:(On (Tarski-Class ((g \/ a) \/ omega))),(On (Tarski-Class ((g \/ a) \/ omega))):]
dom b is epsilon-transitive epsilon-connected ordinal non empty () set
((Tarski-Class ((g \/ U) \/ omega)),(On (Tarski-Class ((g \/ U) \/ omega))),a,a) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class ((g \/ U) \/ omega)
((Tarski-Class ((g \/ U) \/ omega)),(On (Tarski-Class ((g \/ U) \/ omega))),a,U) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class ((g \/ U) \/ omega)
((Tarski-Class ((g \/ a) \/ omega)),(On (Tarski-Class ((g \/ a) \/ omega))),b,U) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class ((g \/ a) \/ omega)
((Tarski-Class ((g \/ a) \/ omega)),(On (Tarski-Class ((g \/ a) \/ omega))),b,a) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class ((g \/ a) \/ omega)
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
(g,a) is epsilon-transitive epsilon-connected ordinal () set
g \/ a is epsilon-transitive epsilon-connected ordinal () set
(g \/ a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((g \/ a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((g \/ a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((g \/ a) \/ omega))) . g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((g \/ a) \/ omega))) . g) . a is epsilon-transitive epsilon-connected ordinal () set
(g,U) is epsilon-transitive epsilon-connected ordinal () set
g \/ U is epsilon-transitive epsilon-connected ordinal () set
(g \/ U) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((g \/ U) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((g \/ U) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((g \/ U) \/ omega))) . g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((g \/ U) \/ omega))) . g) . U is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
U is epsilon-transitive epsilon-connected ordinal () set
(a,U) is epsilon-transitive epsilon-connected ordinal () set
a \/ U is epsilon-transitive epsilon-connected ordinal () set
(a \/ U) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((a \/ U) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((a \/ U) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((a \/ U) \/ omega))) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((a \/ U) \/ omega))) . a) . U is epsilon-transitive epsilon-connected ordinal () set
g is epsilon-transitive epsilon-connected ordinal () set
o is epsilon-transitive epsilon-connected ordinal () set
(g,o) is epsilon-transitive epsilon-connected ordinal () set
g \/ o is epsilon-transitive epsilon-connected ordinal () set
(g \/ o) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((g \/ o) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((g \/ o) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((g \/ o) \/ omega))) . g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((g \/ o) \/ omega))) . g) . o is epsilon-transitive epsilon-connected ordinal () set
(a,(g,o)) is epsilon-transitive epsilon-connected ordinal () set
a \/ (g,o) is epsilon-transitive epsilon-connected ordinal () set
(a \/ (g,o)) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((a \/ (g,o)) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((a \/ (g,o)) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((a \/ (g,o)) \/ omega))) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((a \/ (g,o)) \/ omega))) . a) . (g,o) is epsilon-transitive epsilon-connected ordinal () set
(g,(a,U)) is epsilon-transitive epsilon-connected ordinal () set
g \/ (a,U) is epsilon-transitive epsilon-connected ordinal () set
(g \/ (a,U)) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((g \/ (a,U)) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((g \/ (a,U)) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((g \/ (a,U)) \/ omega))) . g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((g \/ (a,U)) \/ omega))) . g) . (a,U) is epsilon-transitive epsilon-connected ordinal () set
(a,(g,o)) is epsilon-transitive epsilon-connected ordinal () set
a \/ (g,o) is epsilon-transitive epsilon-connected ordinal () set
(a \/ (g,o)) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((a \/ (g,o)) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((a \/ (g,o)) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((a \/ (g,o)) \/ omega))) . a is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((a \/ (g,o)) \/ omega))) . a) . (g,o) is epsilon-transitive epsilon-connected ordinal () set
(g,(a,U)) is epsilon-transitive epsilon-connected ordinal () set
g \/ (a,U) is epsilon-transitive epsilon-connected ordinal () set
(g \/ (a,U)) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((g \/ (a,U)) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((g \/ (a,U)) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((g \/ (a,U)) \/ omega))) . g is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((g \/ (a,U)) \/ omega))) . g) . (a,U) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive non empty subset-closed Tarski non countable universal set
(a) is Relation-like Function-like T-Sequence-like () set
(a) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(a,omega) is Relation-like On a -defined On a -valued Function-like T-Sequence-like non empty V26( On a) V30( On a, On a) Ordinal-yielding Element of bool [:(On a),(On a):]
On a is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On a),(On a):] is Relation-like non empty set
bool [:(On a),(On a):] is non empty set
((a,omega)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom (a,omega) is epsilon-transitive epsilon-connected ordinal non empty () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a,omega) : b₁ is_a_fixpoint_of (a,omega) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a,omega) : b₁ is_a_fixpoint_of (a,omega) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a,omega) : b₁ is_a_fixpoint_of (a,omega) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a,omega) : b₁ is_a_fixpoint_of (a,omega) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a,omega) : b₁ is_a_fixpoint_of (a,omega) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a,omega) : b₁ is_a_fixpoint_of (a,omega) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a,omega) : b₁ is_a_fixpoint_of (a,omega) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a,omega) : b₁ is_a_fixpoint_of (a,omega) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom (a,omega) : b₁ is_a_fixpoint_of (a,omega) } ))) is Relation-like Function-like set
0-element_of a is epsilon-transitive epsilon-connected ordinal () Element of a
succ (0-element_of a) is epsilon-transitive epsilon-connected ordinal non empty () Element of a
{(0-element_of a)} is non empty trivial finite 1 -element set
(0-element_of a) \/ {(0-element_of a)} is non empty set
(a) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((a) . 0)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom ((a) . 0) is epsilon-transitive epsilon-connected ordinal () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((a) . 0) : b₁ is_a_fixpoint_of (a) . 0 } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((a) . 0) : b₁ is_a_fixpoint_of (a) . 0 } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((a) . 0) : b₁ is_a_fixpoint_of (a) . 0 } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((a) . 0) : b₁ is_a_fixpoint_of (a) . 0 } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((a) . 0) : b₁ is_a_fixpoint_of (a) . 0 } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((a) . 0) : b₁ is_a_fixpoint_of (a) . 0 } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((a) . 0) : b₁ is_a_fixpoint_of (a) . 0 } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((a) . 0) : b₁ is_a_fixpoint_of (a) . 0 } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((a) . 0) : b₁ is_a_fixpoint_of (a) . 0 } ))) is Relation-like Function-like set
a is epsilon-transitive epsilon-connected ordinal () set
(1,a) is epsilon-transitive epsilon-connected ordinal () set
1 \/ a is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ a) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ a) \/ omega))) . 1) . a is epsilon-transitive epsilon-connected ordinal () set
a \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (a \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (a \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (a \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (a \/ omega))) . 1) . a is epsilon-transitive epsilon-connected ordinal () set
(0,(1,a)) is epsilon-transitive epsilon-connected ordinal () set
0 \/ (1,a) is epsilon-transitive epsilon-connected ordinal () set
(0 \/ (1,a)) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((0 \/ (1,a)) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((0 \/ (1,a)) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((0 \/ (1,a)) \/ omega))) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((0 \/ (1,a)) \/ omega))) . 0) . (1,a) is epsilon-transitive epsilon-connected ordinal () set
(1,a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1,a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1,a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1,a) \/ omega))) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1,a) \/ omega))) . 0) . (1,a) is epsilon-transitive epsilon-connected ordinal () set
succ 0 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () Element of omega
{0} is functional non empty trivial finite finite-membered 1 -element with_common_domain set
0 \/ {0} is non empty finite finite-membered set
((succ 0),a) is epsilon-transitive epsilon-connected ordinal () set
(succ 0) \/ a is epsilon-transitive epsilon-connected ordinal non empty () set
((succ 0) \/ a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (((succ 0) \/ a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (((succ 0) \/ a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (((succ 0) \/ a) \/ omega))) . (succ 0) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (((succ 0) \/ a) \/ omega))) . (succ 0)) . a is epsilon-transitive epsilon-connected ordinal () set
((Tarski-Class (a \/ omega))) . (succ 0) is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (a \/ omega))) . (succ 0)) . a is epsilon-transitive epsilon-connected ordinal () set
exp (omega,(1,a)) is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
a \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (a \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
0-element_of (Tarski-Class (a \/ omega)) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class (a \/ omega)
1-element_of (Tarski-Class (a \/ omega)) is epsilon-transitive epsilon-connected ordinal non empty () Element of Tarski-Class (a \/ omega)
On (Tarski-Class (a \/ omega)) is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On (Tarski-Class (a \/ omega))),(On (Tarski-Class (a \/ omega))):] is Relation-like non empty set
bool [:(On (Tarski-Class (a \/ omega))),(On (Tarski-Class (a \/ omega))):] is non empty set
((Tarski-Class (a \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (a \/ omega))) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
((Tarski-Class (a \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
o is Relation-like On (Tarski-Class (a \/ omega)) -defined On (Tarski-Class (a \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class (a \/ omega))) V30( On (Tarski-Class (a \/ omega)), On (Tarski-Class (a \/ omega))) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On (Tarski-Class (a \/ omega))),(On (Tarski-Class (a \/ omega))):]
((Tarski-Class (a \/ omega)),omega) is Relation-like On (Tarski-Class (a \/ omega)) -defined On (Tarski-Class (a \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class (a \/ omega))) V30( On (Tarski-Class (a \/ omega)), On (Tarski-Class (a \/ omega))) Ordinal-yielding Element of bool [:(On (Tarski-Class (a \/ omega))),(On (Tarski-Class (a \/ omega))):]
(((Tarski-Class (a \/ omega)),omega)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
dom ((Tarski-Class (a \/ omega)),omega) is epsilon-transitive epsilon-connected ordinal non empty () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),omega) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),omega) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),omega) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),omega) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),omega) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),omega) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),omega) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),omega) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),omega) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),omega) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),omega) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),omega) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),omega) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),omega) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),omega) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),omega) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),omega) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),omega) } ))) is Relation-like Function-like set
g is Relation-like On (Tarski-Class (a \/ omega)) -defined On (Tarski-Class (a \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class (a \/ omega))) V30( On (Tarski-Class (a \/ omega)), On (Tarski-Class (a \/ omega))) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On (Tarski-Class (a \/ omega))),(On (Tarski-Class (a \/ omega))):]
(g) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing continuous non-decreasing () set
dom g is epsilon-transitive epsilon-connected ordinal non empty () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom g : b₁ is_a_fixpoint_of g } ))) is Relation-like Function-like set
((Tarski-Class (a \/ omega)),(On (Tarski-Class (a \/ omega))),g,a) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class (a \/ omega)
(0,a) is epsilon-transitive epsilon-connected ordinal () set
0 \/ a is epsilon-transitive epsilon-connected ordinal non empty () set
(0 \/ a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((0 \/ a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((0 \/ a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((0 \/ a) \/ omega))) . 0 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((0 \/ a) \/ omega))) . 0) . a is epsilon-transitive epsilon-connected ordinal () set
(((Tarski-Class (a \/ omega))) . 0) . a is epsilon-transitive epsilon-connected ordinal () set
exp (omega,a) is epsilon-transitive epsilon-connected ordinal () set
dom (g) is epsilon-transitive epsilon-connected ordinal () set
f is epsilon-transitive epsilon-connected ordinal () set
(g) . f is epsilon-transitive epsilon-connected ordinal () set
(1,f) is epsilon-transitive epsilon-connected ordinal () set
1 \/ f is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ f) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ f) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ f) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ f) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ f) \/ omega))) . 1) . f is epsilon-transitive epsilon-connected ordinal () set
f \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (f \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (f \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (f \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (f \/ omega))) . 1) . f is epsilon-transitive epsilon-connected ordinal () set
1-element_of (Tarski-Class (f \/ omega)) is epsilon-transitive epsilon-connected ordinal non empty () Element of Tarski-Class (f \/ omega)
(1,0) is epsilon-transitive epsilon-connected ordinal () set
1 \/ 0 is epsilon-transitive epsilon-connected ordinal natural non empty finite cardinal () set
(1 \/ 0) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ 0) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ 0) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ 0) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ 0) \/ omega))) . 1) . 0 is epsilon-transitive epsilon-connected ordinal () set
0 \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (0 \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (0 \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (0 \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (0 \/ omega))) . 1) . 0 is epsilon-transitive epsilon-connected ordinal () set
epsilon_ 0 is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
a is epsilon-transitive epsilon-connected ordinal () set
epsilon_ a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
U is epsilon-transitive epsilon-connected ordinal () set
(1,U) is epsilon-transitive epsilon-connected ordinal () set
1 \/ U is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ U) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ U) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ U) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ U) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ U) \/ omega))) . 1) . U is epsilon-transitive epsilon-connected ordinal () set
U \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (U \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (U \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (U \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (U \/ omega))) . 1) . U is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
(1,a) is epsilon-transitive epsilon-connected ordinal () set
1 \/ a is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ a) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ a) \/ omega))) . 1) . a is epsilon-transitive epsilon-connected ordinal () set
a \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (a \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (a \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (a \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (a \/ omega))) . 1) . a is epsilon-transitive epsilon-connected ordinal () set
epsilon_ a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
succ a is epsilon-transitive epsilon-connected ordinal non empty () set
{a} is non empty trivial finite 1 -element set
a \/ {a} is non empty set
(1,(succ a)) is epsilon-transitive epsilon-connected ordinal () set
1 \/ (succ a) is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ (succ a)) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ (succ a)) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ (succ a)) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ (succ a)) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ (succ a)) \/ omega))) . 1) . (succ a) is epsilon-transitive epsilon-connected ordinal () set
(succ a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((succ a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((succ a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((succ a) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((succ a) \/ omega))) . 1) . (succ a) is epsilon-transitive epsilon-connected ordinal () set
epsilon_ (succ a) is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
U is epsilon-transitive epsilon-connected ordinal () set
epsilon_ U is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
g is epsilon-transitive epsilon-connected ordinal () set
(1,g) is epsilon-transitive epsilon-connected ordinal () set
1 \/ g is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ g) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ g) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ g) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ g) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ g) \/ omega))) . 1) . g is epsilon-transitive epsilon-connected ordinal () set
g \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (g \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (g \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (g \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (g \/ omega))) . 1) . g is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite () set
(1,a) is epsilon-transitive epsilon-connected ordinal () set
1 \/ a is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ a) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ a) \/ omega))) . 1) . a is epsilon-transitive epsilon-connected ordinal () set
a \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (a \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (a \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (a \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (a \/ omega))) . 1) . a is epsilon-transitive epsilon-connected ordinal () set
epsilon_ a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
1-element_of (Tarski-Class (a \/ omega)) is epsilon-transitive epsilon-connected ordinal non empty () Element of Tarski-Class (a \/ omega)
On (Tarski-Class (a \/ omega)) is epsilon-transitive epsilon-connected ordinal non empty () set
[:(On (Tarski-Class (a \/ omega))),(On (Tarski-Class (a \/ omega))):] is Relation-like non empty set
bool [:(On (Tarski-Class (a \/ omega))),(On (Tarski-Class (a \/ omega))):] is non empty set
o is epsilon-transitive epsilon-connected ordinal non trivial () Element of Tarski-Class (a \/ omega)
((Tarski-Class (a \/ omega)),o) is Relation-like On (Tarski-Class (a \/ omega)) -defined On (Tarski-Class (a \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class (a \/ omega))) V30( On (Tarski-Class (a \/ omega)), On (Tarski-Class (a \/ omega))) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On (Tarski-Class (a \/ omega))),(On (Tarski-Class (a \/ omega))):]
g is Relation-like On (Tarski-Class (a \/ omega)) -defined On (Tarski-Class (a \/ omega)) -valued Function-like T-Sequence-like non empty V26( On (Tarski-Class (a \/ omega))) V30( On (Tarski-Class (a \/ omega)), On (Tarski-Class (a \/ omega))) Ordinal-yielding increasing continuous non-decreasing () Element of bool [:(On (Tarski-Class (a \/ omega))),(On (Tarski-Class (a \/ omega))):]
(((Tarski-Class (a \/ omega)),o)) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing continuous non-decreasing () set
dom ((Tarski-Class (a \/ omega)),o) is epsilon-transitive epsilon-connected ordinal non empty () set
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),o) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),o) } is set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),o) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),o) } ) is Relation-like Function-like T-Sequence-like Ordinal-yielding increasing non-decreasing set
( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),o) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),o) } ) is epsilon-transitive epsilon-connected ordinal () set
On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),o) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),o) } is () set
RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),o) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),o) } ) is Relation-like well-ordering V34() V37() V41() set
order_type_of (RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),o) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),o) } )) is epsilon-transitive epsilon-connected ordinal () set
RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),o) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),o) } ) is Relation-like well_founded well-ordering V34() V37() V39() V41() set
canonical_isomorphism_of ((RelIncl ( { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),o) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),o) } )),(RelIncl (On { b₁ where b₁ is epsilon-transitive epsilon-connected ordinal () Element of dom ((Tarski-Class (a \/ omega)),o) : b₁ is_a_fixpoint_of ((Tarski-Class (a \/ omega)),o) } ))) is Relation-like Function-like set
dom g is epsilon-transitive epsilon-connected ordinal non empty () set
((Tarski-Class (a \/ omega)),(On (Tarski-Class (a \/ omega))),g,a) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class (a \/ omega)
g | a is Relation-like a -defined On (Tarski-Class (a \/ omega)) -defined On (Tarski-Class (a \/ omega)) -valued rng g -valued Function-like T-Sequence-like Ordinal-yielding Element of bool [:(On (Tarski-Class (a \/ omega))),(On (Tarski-Class (a \/ omega))):]
rng g is non empty () set
Union (g | a) is epsilon-transitive epsilon-connected ordinal () set
rng (g | a) is () set
union (rng (g | a)) is epsilon-transitive epsilon-connected ordinal () set
dom (g | a) is epsilon-transitive epsilon-connected ordinal () set
a is set
b is set
(g | a) . b is set
y is epsilon-transitive epsilon-connected ordinal () set
((Tarski-Class (a \/ omega)),(On (Tarski-Class (a \/ omega))),g,y) is epsilon-transitive epsilon-connected ordinal () Element of Tarski-Class (a \/ omega)
(1,y) is epsilon-transitive epsilon-connected ordinal () set
1 \/ y is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ y) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ y) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ y) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ y) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ y) \/ omega))) . 1) . y is epsilon-transitive epsilon-connected ordinal () set
y \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (y \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (y \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (y \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (y \/ omega))) . 1) . y is epsilon-transitive epsilon-connected ordinal () set
epsilon_ y is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
a is epsilon-transitive epsilon-connected ordinal () set
b is epsilon-transitive epsilon-connected ordinal () set
epsilon_ b is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
(1,b) is epsilon-transitive epsilon-connected ordinal () set
1 \/ b is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ b) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ b) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ b) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ b) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ b) \/ omega))) . 1) . b is epsilon-transitive epsilon-connected ordinal () set
b \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (b \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (b \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (b \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (b \/ omega))) . 1) . b is epsilon-transitive epsilon-connected ordinal () set
a is epsilon-transitive epsilon-connected ordinal () set
(1,a) is epsilon-transitive epsilon-connected ordinal () set
1 \/ a is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ a) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ a) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ a) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ a) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ a) \/ omega))) . 1) . a is epsilon-transitive epsilon-connected ordinal () set
a \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (a \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (a \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (a \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (a \/ omega))) . 1) . a is epsilon-transitive epsilon-connected ordinal () set
epsilon_ a is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
g is epsilon-transitive epsilon-connected ordinal () set
(1,g) is epsilon-transitive epsilon-connected ordinal () set
1 \/ g is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ g) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ g) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ g) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ g) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ g) \/ omega))) . 1) . g is epsilon-transitive epsilon-connected ordinal () set
g \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (g \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (g \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (g \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (g \/ omega))) . 1) . g is epsilon-transitive epsilon-connected ordinal () set
epsilon_ g is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
succ g is epsilon-transitive epsilon-connected ordinal non empty () set
{g} is non empty trivial finite 1 -element set
g \/ {g} is non empty set
(1,(succ g)) is epsilon-transitive epsilon-connected ordinal () set
1 \/ (succ g) is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ (succ g)) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ (succ g)) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ (succ g)) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ (succ g)) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ (succ g)) \/ omega))) . 1) . (succ g) is epsilon-transitive epsilon-connected ordinal () set
(succ g) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((succ g) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((succ g) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((succ g) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((succ g) \/ omega))) . 1) . (succ g) is epsilon-transitive epsilon-connected ordinal () set
epsilon_ (succ g) is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set
g is epsilon-transitive epsilon-connected ordinal () set
(1,g) is epsilon-transitive epsilon-connected ordinal () set
1 \/ g is epsilon-transitive epsilon-connected ordinal non empty () set
(1 \/ g) \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class ((1 \/ g) \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class ((1 \/ g) \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class ((1 \/ g) \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class ((1 \/ g) \/ omega))) . 1) . g is epsilon-transitive epsilon-connected ordinal () set
g \/ omega is epsilon-transitive epsilon-connected ordinal non empty () set
Tarski-Class (g \/ omega) is epsilon-transitive non empty subset-closed Tarski non countable universal set
((Tarski-Class (g \/ omega))) is Relation-like Function-like T-Sequence-like () set
((Tarski-Class (g \/ omega))) . 1 is Relation-like Function-like T-Sequence-like Ordinal-yielding set
(((Tarski-Class (g \/ omega))) . 1) . g is epsilon-transitive epsilon-connected ordinal () set
epsilon_ g is epsilon-transitive epsilon-connected ordinal limit_ordinal non empty non trivial non finite epsilon () set