:: Universal Classes
:: by Bogdan Nowak and Grzegorz Bancerek
::
:: Received April 10, 1990
:: Copyright (c) 1990-2012 Association of Mizar Users


begin

Lm1: for X being set holds Tarski-Class X is Tarski
proof end;

registration
cluster Tarski -> subset-closed for set ;
coherence
for b1 being set st b1 is Tarski holds
b1 is subset-closed by CLASSES1:def 2;
end;

registration
let X be set ;
cluster Tarski-Class X -> Tarski ;
coherence
Tarski-Class X is Tarski by Lm1;
end;

theorem Th1: :: CLASSES2:1
for W, X being set st W is subset-closed & X in W holds
( not X,W are_equipotent & card X in card W )
proof end;

theorem Th2: :: CLASSES2:2
for W, x, y being set st W is Tarski & x in W & y in W holds
( {x} in W & {x,y} in W )
proof end;

theorem Th3: :: CLASSES2:3
for W, x, y being set st W is Tarski & x in W & y in W holds
[x,y] in W
proof end;

theorem Th4: :: CLASSES2:4
for W, X being set st W is Tarski & X in W holds
Tarski-Class X c= W
proof end;

theorem Th5: :: CLASSES2:5
for A being Ordinal
for W being set st W is Tarski & A in W holds
( succ A in W & A c= W )
proof end;

theorem :: CLASSES2:6
for A being Ordinal
for W being set st A in Tarski-Class W holds
( succ A in Tarski-Class W & A c= Tarski-Class W ) by Th5;

theorem Th7: :: CLASSES2:7
for W, X being set st W is subset-closed & X is epsilon-transitive & X in W holds
X c= W
proof end;

theorem :: CLASSES2:8
for X, W being set st X is epsilon-transitive & X in Tarski-Class W holds
X c= Tarski-Class W by Th7;

theorem Th9: :: CLASSES2:9
for W being set st W is Tarski holds
On W = card W
proof end;

theorem :: CLASSES2:10
for W being set holds On (Tarski-Class W) = card (Tarski-Class W) by Th9;

theorem Th11: :: CLASSES2:11
for W, X being set st W is Tarski & X in W holds
card X in W
proof end;

theorem :: CLASSES2:12
for X, W being set st X in Tarski-Class W holds
card X in Tarski-Class W by Th11;

theorem Th13: :: CLASSES2:13
for W, x being set st W is Tarski & x in card W holds
x in W
proof end;

theorem :: CLASSES2:14
for x, W being set st x in card (Tarski-Class W) holds
x in Tarski-Class W by Th13;

theorem :: CLASSES2:15
for m being Cardinal
for W being set st W is Tarski & m in card W holds
m in W
proof end;

theorem :: CLASSES2:16
for m being Cardinal
for W being set st m in card (Tarski-Class W) holds
m in Tarski-Class W
proof end;

theorem :: CLASSES2:17
for m being Cardinal
for W being set st W is Tarski & m in W holds
m c= W by Th5;

theorem :: CLASSES2:18
for m being Cardinal
for W being set st m in Tarski-Class W holds
m c= Tarski-Class W by Th5;

theorem Th19: :: CLASSES2:19
for W being set st W is Tarski holds
card W is limit_ordinal
proof end;

theorem :: CLASSES2:20
for W being set st W is Tarski & W <> {} holds
( card W <> 0 & card W <> {} & card W is limit_ordinal ) by Th19;

theorem Th21: :: CLASSES2:21
for W being set holds
( card (Tarski-Class W) <> 0 & card (Tarski-Class W) <> {} & card (Tarski-Class W) is limit_ordinal )
proof end;

theorem Th22: :: CLASSES2:22
for W, X being set st W is Tarski & ( ( X in W & W is epsilon-transitive ) or ( X in W & X c= W ) or ( card X in card W & X c= W ) ) holds
Funcs (X,W) c= W
proof end;

theorem :: CLASSES2:23
for X, W being set st ( ( X in Tarski-Class W & W is epsilon-transitive ) or ( X in Tarski-Class W & X c= Tarski-Class W ) or ( card X in card (Tarski-Class W) & X c= Tarski-Class W ) ) holds
Funcs (X,(Tarski-Class W)) c= Tarski-Class W
proof end;

theorem Th24: :: CLASSES2:24
for L being T-Sequence st dom L is limit_ordinal & ( for A being Ordinal st A in dom L holds
L . A = Rank A ) holds
Rank (dom L) = Union L
proof end;

defpred S1[ Ordinal] means for W being set st W is Tarski & $1 in On W holds
( card (Rank $1) in card W & Rank $1 in W );

Lm2: S1[ {} ]
by Th9, CARD_1:27, CLASSES1:29, ORDINAL1:def 9;

Lm3: for A being Ordinal st S1[A] holds
S1[ succ A]
proof end;

Lm4: for A being Ordinal st A <> {} & A is limit_ordinal & ( for B being Ordinal st B in A holds
S1[B] ) holds
S1[A]
proof end;

theorem Th25: :: CLASSES2:25
for A being Ordinal
for W being set st W is Tarski & A in On W holds
( card (Rank A) in card W & Rank A in W )
proof end;

theorem :: CLASSES2:26
for A being Ordinal
for W being set st A in On (Tarski-Class W) holds
( card (Rank A) in card (Tarski-Class W) & Rank A in Tarski-Class W ) by Th25;

theorem Th27: :: CLASSES2:27
for W being set st W is Tarski holds
Rank (card W) c= W
proof end;

theorem Th28: :: CLASSES2:28
for W being set holds Rank (card (Tarski-Class W)) c= Tarski-Class W
proof end;

deffunc H1( set ) -> set = the_rank_of $1;

deffunc H2( set ) -> set = card (bool $1);

theorem Th29: :: CLASSES2:29
for W, X being set st W is Tarski & W is epsilon-transitive & X in W holds
the_rank_of X in W
proof end;

theorem Th30: :: CLASSES2:30
for W being set st W is Tarski & W is epsilon-transitive holds
W c= Rank (card W)
proof end;

theorem Th31: :: CLASSES2:31
for W being set st W is Tarski & W is epsilon-transitive holds
Rank (card W) = W
proof end;

theorem :: CLASSES2:32
for A being Ordinal
for W being set st W is Tarski & A in On W holds
card (Rank A) c= card W
proof end;

theorem :: CLASSES2:33
for A being Ordinal
for W being set st A in On (Tarski-Class W) holds
card (Rank A) c= card (Tarski-Class W)
proof end;

theorem Th34: :: CLASSES2:34
for W being set st W is Tarski holds
card W = card (Rank (card W))
proof end;

theorem :: CLASSES2:35
for W being set holds card (Tarski-Class W) = card (Rank (card (Tarski-Class W))) by Th34;

theorem Th36: :: CLASSES2:36
for W, X being set st W is Tarski & X c= Rank (card W) & not X, Rank (card W) are_equipotent holds
X in Rank (card W)
proof end;

theorem :: CLASSES2:37
for X, W being set holds
( not X c= Rank (card (Tarski-Class W)) or X, Rank (card (Tarski-Class W)) are_equipotent or X in Rank (card (Tarski-Class W)) ) by Th36;

theorem Th38: :: CLASSES2:38
for W being set st W is Tarski holds
Rank (card W) is Tarski
proof end;

theorem :: CLASSES2:39
for W being set holds Rank (card (Tarski-Class W)) is Tarski by Th38;

theorem Th40: :: CLASSES2:40
for A being Ordinal
for X being set st X is epsilon-transitive & A in the_rank_of X holds
ex Y being set st
( Y in X & the_rank_of Y = A )
proof end;

theorem Th41: :: CLASSES2:41
for X being set st X is epsilon-transitive holds
card (the_rank_of X) c= card X
proof end;

theorem Th42: :: CLASSES2:42
for W, X being set st W is Tarski & X is epsilon-transitive & X in W holds
X in Rank (card W)
proof end;

theorem :: CLASSES2:43
for X, W being set st X is epsilon-transitive & X in Tarski-Class W holds
X in Rank (card (Tarski-Class W)) by Th42;

theorem Th44: :: CLASSES2:44
for W being set st W is epsilon-transitive holds
Rank (card (Tarski-Class W)) is_Tarski-Class_of W
proof end;

theorem :: CLASSES2:45
for W being set st W is epsilon-transitive holds
Rank (card (Tarski-Class W)) = Tarski-Class W
proof end;

definition
let IT be set ;
attr IT is universal means :Def1: :: CLASSES2:def 1
( IT is epsilon-transitive & IT is Tarski );
end;

:: deftheorem Def1 defines universal CLASSES2:def 1 :
for IT being set holds
( IT is universal iff ( IT is epsilon-transitive & IT is Tarski ) );

registration
cluster universal -> epsilon-transitive Tarski for set ;
coherence
for b1 being set st b1 is universal holds
( b1 is epsilon-transitive & b1 is Tarski ) by Def1;
cluster epsilon-transitive Tarski -> universal for set ;
coherence
for b1 being set st b1 is epsilon-transitive & b1 is Tarski holds
b1 is universal by Def1;
end;

registration
cluster non empty universal for set ;
existence
ex b1 being set st
( b1 is universal & not b1 is empty )
proof end;
end;

definition
mode Universe is non empty universal set ;
end;

theorem Th46: :: CLASSES2:46
for U being Universe holds On U is Ordinal
proof end;

theorem Th47: :: CLASSES2:47
for X being set st X is epsilon-transitive holds
Tarski-Class X is universal
proof end;

theorem :: CLASSES2:48
for U being Universe holds Tarski-Class U is Universe by Th47;

registration
let U be Universe;
cluster On U -> ordinal ;
coherence
On U is ordinal by Th46;
cluster Tarski-Class U -> universal ;
coherence
Tarski-Class U is universal by Th47;
end;

theorem Th49: :: CLASSES2:49
for A being Ordinal holds Tarski-Class A is universal
proof end;

registration
let A be Ordinal;
cluster Tarski-Class A -> universal ;
coherence
Tarski-Class A is universal by Th49;
end;

theorem Th50: :: CLASSES2:50
for U being Universe holds U = Rank (On U)
proof end;

theorem Th51: :: CLASSES2:51
for U being Universe holds
( On U <> {} & On U is limit_ordinal )
proof end;

theorem :: CLASSES2:52
for U1, U2 being Universe holds
( U1 in U2 or U1 = U2 or U2 in U1 )
proof end;

theorem Th53: :: CLASSES2:53
for U1, U2 being Universe holds
( U1 c= U2 or U2 in U1 )
proof end;

theorem Th54: :: CLASSES2:54
for U1, U2 being Universe holds U1,U2 are_c=-comparable
proof end;

theorem :: CLASSES2:55
for U1, U2 being Universe holds
( U1 \/ U2 is Universe & U1 /\ U2 is Universe )
proof end;

theorem Th56: :: CLASSES2:56
for U being Universe holds {} in U
proof end;

theorem :: CLASSES2:57
for x being set
for U being Universe st x in U holds
{x} in U by Th2;

theorem :: CLASSES2:58
for x, y being set
for U being Universe st x in U & y in U holds
( {x,y} in U & [x,y] in U ) by Th2, Th3;

theorem Th59: :: CLASSES2:59
for X being set
for U being Universe st X in U holds
( bool X in U & union X in U & meet X in U )
proof end;

theorem Th60: :: CLASSES2:60
for X, Y being set
for U being Universe st X in U & Y in U holds
( X \/ Y in U & X /\ Y in U & X \ Y in U & X \+\ Y in U )
proof end;

theorem Th61: :: CLASSES2:61
for X, Y being set
for U being Universe st X in U & Y in U holds
( [:X,Y:] in U & Funcs (X,Y) in U )
proof end;

registration
let U1 be Universe;
cluster non empty for Element of U1;
existence
not for b1 being Element of U1 holds b1 is empty
proof end;
end;

definition
let U be Universe;
let u be Element of U;
:: original: {
redefine func {u} -> Element of U;
coherence
{u} is Element of U by Th2;
:: original: bool
redefine func bool u -> Element of U;
coherence
bool u is Element of U by Th59;
:: original: union
redefine func union u -> Element of U;
coherence
union u is Element of U by Th59;
:: original: meet
redefine func meet u -> Element of U;
coherence
meet u is Element of U by Th59;
let v be Element of U;
:: original: {
redefine func {u,v} -> Element of U;
coherence
{u,v} is Element of U by Th2;
:: original: [
redefine func [u,v] -> Element of U;
coherence
[u,v] is Element of U by Th3;
:: original: \/
redefine func u \/ v -> Element of U;
coherence
u \/ v is Element of U by Th60;
:: original: /\
redefine func u /\ v -> Element of U;
coherence
u /\ v is Element of U by Th60;
:: original: \
redefine func u \ v -> Element of U;
coherence
u \ v is Element of U by Th60;
:: original: \+\
redefine func u \+\ v -> Element of U;
coherence
u \+\ v is Element of U by Th60;
:: original: [:
redefine func [:u,v:] -> Element of U;
coherence
[:u,v:] is Element of U by Th61;
:: original: Funcs
redefine func Funcs (u,v) -> Element of U;
coherence
Funcs (u,v) is Element of U by Th61;
end;

definition
func FinSETS -> Universe equals :: CLASSES2:def 2
 Tarski-Class {};
correctness
coherence
Tarski-Class {} is Universe;
;
end;

:: deftheorem defines FinSETS CLASSES2:def 2 :
FinSETS = Tarski-Class {};

Lm5: omega is limit_ordinal
by ORDINAL1:def 11;

theorem Th62: :: CLASSES2:62
card (Rank omega) = card omega
proof end;

theorem Th63: :: CLASSES2:63
Rank omega is Tarski
proof end;

theorem :: CLASSES2:64
FinSETS = Rank omega
proof end;

definition
func SETS -> Universe equals :: CLASSES2:def 3
 Tarski-Class FinSETS;
correctness
coherence
Tarski-Class FinSETS is Universe;
;
end;

:: deftheorem defines SETS CLASSES2:def 3 :
SETS = Tarski-Class FinSETS;

registration
let X be set ;
cluster the_transitive-closure_of X -> epsilon-transitive ;
coherence
the_transitive-closure_of X is epsilon-transitive by CLASSES1:51;
end;

registration
let X be epsilon-transitive set ;
cluster Tarski-Class X -> epsilon-transitive ;
coherence
Tarski-Class X is epsilon-transitive by CLASSES1:23;
end;

definition
let X be set ;
func Universe_closure X -> Universe means :Def4: :: CLASSES2:def 4
( X c= it & ( for Y being Universe st X c= Y holds
it c= Y ) );
uniqueness
for b1, b2 being Universe st X c= b1 & ( for Y being Universe st X c= Y holds
b1 c= Y ) & X c= b2 & ( for Y being Universe st X c= Y holds
b2 c= Y ) holds
b1 = b2
proof end;
existence
ex b1 being Universe st
( X c= b1 & ( for Y being Universe st X c= Y holds
b1 c= Y ) )
proof end;
end;

:: deftheorem Def4 defines Universe_closure CLASSES2:def 4 :
for X being set
for b2 being Universe holds
( b2 = Universe_closure X iff ( X c= b2 & ( for Y being Universe st X c= Y holds
b2 c= Y ) ) );

deffunc H3( Ordinal, set ) -> set = Tarski-Class $2;

deffunc H4( Ordinal, T-Sequence) -> Universe = Universe_closure (Union $2);

definition
mode FinSet is Element of FinSETS ;
mode Set is Element of SETS ;
let A be Ordinal;
func UNIVERSE A -> set means :Def5: :: CLASSES2:def 5
 ex L being T-Sequence st
( it = last L & dom L = succ A & L . {} = FinSETS & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = Tarski-Class (L . C) ) & ( for C being Ordinal st C in succ A & C <> {} & C is limit_ordinal holds
L . C = Universe_closure (Union (L | C)) ) );
correctness
existence
ex b1 being set ex L being T-Sequence st
( b1 = last L & dom L = succ A & L . {} = FinSETS & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = Tarski-Class (L . C) ) & ( for C being Ordinal st C in succ A & C <> {} & C is limit_ordinal holds
L . C = Universe_closure (Union (L | C)) ) );
uniqueness
for b1, b2 being set st ex L being T-Sequence st
( b1 = last L & dom L = succ A & L . {} = FinSETS & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = Tarski-Class (L . C) ) & ( for C being Ordinal st C in succ A & C <> {} & C is limit_ordinal holds
L . C = Universe_closure (Union (L | C)) ) ) & ex L being T-Sequence st
( b2 = last L & dom L = succ A & L . {} = FinSETS & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = Tarski-Class (L . C) ) & ( for C being Ordinal st C in succ A & C <> {} & C is limit_ordinal holds
L . C = Universe_closure (Union (L | C)) ) ) holds
b1 = b2;
proof end;
end;

:: deftheorem Def5 defines UNIVERSE CLASSES2:def 5 :
for A being Ordinal
for b2 being set holds
( b2 = UNIVERSE A iff ex L being T-Sequence st
( b2 = last L & dom L = succ A & L . {} = FinSETS & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = Tarski-Class (L . C) ) & ( for C being Ordinal st C in succ A & C <> {} & C is limit_ordinal holds
L . C = Universe_closure (Union (L | C)) ) ) );

deffunc H5( Ordinal) -> set = UNIVERSE $1;

Lm6: now :: thesis: ( H5( {} ) = FinSETS & ( for A being Ordinal holds H5( succ A) = H3(A,H5(A)) ) & ( for A being Ordinal
for L being T-Sequence st A <> {} & A is limit_ordinal & dom L = A & ( for B being Ordinal st B in A holds
L . B = H5(B) ) holds
H5(A) = H4(A,L) ) )
A1: for A being Ordinal
for x being set holds
( x = H5(A) iff ex L being T-Sequence st
( x = last L & dom L = succ A & L . {} = FinSETS & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = H3(C,L . C) ) & ( for C being Ordinal st C in succ A & C <> {} & C is limit_ordinal holds
L . C = H4(C,L | C) ) ) ) by Def5;
thus H5( {} ) = FinSETS from ORDINAL2:sch 8(A1); :: thesis: ( ( for A being Ordinal holds H5( succ A) = H3(A,H5(A)) ) & ( for A being Ordinal
for L being T-Sequence st A <> {} & A is limit_ordinal & dom L = A & ( for B being Ordinal st B in A holds
L . B = H5(B) ) holds
H5(A) = H4(A,L) ) )
thus for A being Ordinal holds H5( succ A) = H3(A,H5(A)) from ORDINAL2:sch 9(A1); :: thesis: for A being Ordinal
for L being T-Sequence st A <> {} & A is limit_ordinal & dom L = A & ( for B being Ordinal st B in A holds
L . B = H5(B) ) holds
H5(A) = H4(A,L)
let A be Ordinal; :: thesis: for L being T-Sequence st A <> {} & A is limit_ordinal & dom L = A & ( for B being Ordinal st B in A holds
L . B = H5(B) ) holds
H5(A) = H4(A,L)
let L be T-Sequence; :: thesis: ( A <> {} & A is limit_ordinal & dom L = A & ( for B being Ordinal st B in A holds
L . B = H5(B) ) implies H5(A) = H4(A,L) )
assume that
A2: ( A <> {} & A is limit_ordinal ) and
A3: dom L = A and
A4: for B being Ordinal st B in A holds
L . B = H5(B) ; :: thesis: H5(A) = H4(A,L)
thus H5(A) = H4(A,L) from ORDINAL2:sch 10(A1, A2, A3, A4); :: thesis: verum
end;

registration
let A be Ordinal;
cluster UNIVERSE A -> non empty universal ;
coherence
( UNIVERSE A is universal & not UNIVERSE A is empty )
proof end;
end;

theorem :: CLASSES2:65
UNIVERSE {} = FinSETS by Lm6;

theorem :: CLASSES2:66
for A being Ordinal holds UNIVERSE (succ A) = Tarski-Class (UNIVERSE A) by Lm6;

Lm7: 1 = succ 0
;

theorem :: CLASSES2:67
UNIVERSE 1 = SETS by Lm6, Lm7;

theorem :: CLASSES2:68
for A being Ordinal
for L being T-Sequence st A <> {} & A is limit_ordinal & dom L = A & ( for B being Ordinal st B in A holds
L . B = UNIVERSE B ) holds
UNIVERSE A = Universe_closure (Union L) by Lm6;

theorem :: CLASSES2:69
for U being Universe holds
( FinSETS c= U & Tarski-Class {} c= U & UNIVERSE {} c= U )
proof end;

theorem Th70: :: CLASSES2:70
for A, B being Ordinal holds
( A in B iff UNIVERSE A in UNIVERSE B )
proof end;

theorem :: CLASSES2:71
for A, B being Ordinal st UNIVERSE A = UNIVERSE B holds
A = B
proof end;

theorem :: CLASSES2:72
for A, B being Ordinal holds
( A c= B iff UNIVERSE A c= UNIVERSE B )
proof end;

theorem :: CLASSES2:73
for X being set holds
( Tarski-Class (X,{}) in Tarski-Class (X,1) & Tarski-Class (X,{}) <> Tarski-Class (X,1) )
proof end;