:: Tarski's Classes and Ranks
:: by Grzegorz Bancerek
::
:: Received March 23, 1990
:: Copyright (c) 1990-2012 Association of Mizar Users


begin

definition
let B be set ;
attrB is subset-closed  means :Def1: :: CLASSES1:def 1
 for X, Y being set st X in B & Y c= X holds
Y in B;
end;

:: deftheorem Def1 defines subset-closed CLASSES1:def_1_:_
 for B being set holds
( B is subset-closed iff for X, Y being set st X in B & Y c= X holds
Y in B );

definition
let B be set ;
attrB is Tarski  means :Def2: :: CLASSES1:def 2
( B is subset-closed & ( for X being set st X in B holds
bool X in B ) & ( for X being set holds
( not X c= B or X,B are_equipotent or X in B ) ) );
end;

:: deftheorem Def2 defines Tarski CLASSES1:def_2_:_
 for B being set holds
( B is Tarski iff ( B is subset-closed & ( for X being set st X in B holds
bool X in B ) & ( for X being set holds
( not X c= B or X,B are_equipotent or X in B ) ) ) );

definition
let A, B be set ;
predB is_Tarski-Class_of A means :Def3: :: CLASSES1:def 3
( A in B & B is Tarski );
end;

:: deftheorem Def3 defines is_Tarski-Class_of CLASSES1:def_3_:_
 for A, B being set holds
( B is_Tarski-Class_of A iff ( A in B & B is Tarski ) );

definition
let A be set ;
func Tarski-Class A ->  set  means :Def4: :: CLASSES1:def 4
( it is_Tarski-Class_of A & ( for D being set st D is_Tarski-Class_of A holds
it c= D ) );
existence
 ex b1 being set st
( b1 is_Tarski-Class_of A & ( for D being set st D is_Tarski-Class_of A holds
b1 c= D ) )
proofend;
uniqueness
 for b1, b2 being set st b1 is_Tarski-Class_of A & ( for D being set st D is_Tarski-Class_of A holds
b1 c= D ) & b2 is_Tarski-Class_of A & ( for D being set st D is_Tarski-Class_of A holds
b2 c= D ) holds
b1 = b2
proofend;
end;

:: deftheorem Def4 defines Tarski-Class CLASSES1:def_4_:_
 for A, b2 being set holds
( b2 = Tarski-Class A iff ( b2 is_Tarski-Class_of A & ( for D being set st D is_Tarski-Class_of A holds
b2 c= D ) ) );

registration
let A be set ;
cluster Tarski-Class A ->  non empty ;
coherence
 not Tarski-Class A is empty
proofend;
end;

theorem :: CLASSES1:1
for W being set holds
( W is Tarski iff ( W is subset-closed & ( for X being set st X in W holds
bool X in W ) & ( for X being set st X c= W & card X in card W holds
X in W ) ) )
proofend;

theorem Th2: :: CLASSES1:2
for X being set holds X in Tarski-Class X
proofend;

theorem Th3: :: CLASSES1:3
for Y, X, Z being set st Y in Tarski-Class X & Z c= Y holds
Z in Tarski-Class X
proofend;

theorem Th4: :: CLASSES1:4
for Y, X being set st Y in Tarski-Class X holds
bool Y in Tarski-Class X
proofend;

theorem Th5: :: CLASSES1:5
for Y, X being set holds
( not Y c= Tarski-Class X or Y, Tarski-Class X are_equipotent or Y in Tarski-Class X )
proofend;

theorem :: CLASSES1:6
for Y, X being set st Y c= Tarski-Class X & card Y in card (Tarski-Class X) holds
Y in Tarski-Class X
proofend;

definition
let X be set ;
let A be Ordinal;
func Tarski-Class (X,A) ->  set  means :Def5: :: CLASSES1:def 5
 ex L being T-Sequence st
( it = last L & dom L = succ A & L . {} = {X} & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = ( {  u where u is Element of Tarski-Class X : ex v being Element of Tarski-Class X st
( v in L . C &amp; u c= v )  }  \/  {  (bool v) where v is Element of Tarski-Class X : v in L . C  }  ) \/ ((bool (L . C)) /\ (Tarski-Class X)) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = (union (rng (L | C))) /\ (Tarski-Class X) ) );
correctness
existence
ex b1 being set ex L being T-Sequence st
( b1 = last L &amp; dom L = succ A &amp; L . {} = {X} &amp; ( for C being Ordinal st succ C in succ A holds
L . (succ C) = ( {  u where u is Element of Tarski-Class X : ex v being Element of Tarski-Class X st
( v in L . C &amp; u c= v )  }  \/  {  (bool v) where v is Element of Tarski-Class X : v in L . C  }  ) \/ ((bool (L . C)) /\ (Tarski-Class X)) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = (union (rng (L | C))) /\ (Tarski-Class X) ) );
uniqueness
for b1, b2 being set st ex L being T-Sequence st
( b1 = last L &amp; dom L = succ A &amp; L . {} = {X} &amp; ( for C being Ordinal st succ C in succ A holds
L . (succ C) = ( {  u where u is Element of Tarski-Class X : ex v being Element of Tarski-Class X st
( v in L . C &amp; u c= v )  }  \/  {  (bool v) where v is Element of Tarski-Class X : v in L . C  }  ) \/ ((bool (L . C)) /\ (Tarski-Class X)) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = (union (rng (L | C))) /\ (Tarski-Class X) ) ) &amp; ex L being T-Sequence st
( b2 = last L &amp; dom L = succ A &amp; L . {} = {X} &amp; ( for C being Ordinal st succ C in succ A holds
L . (succ C) = ( {  u where u is Element of Tarski-Class X : ex v being Element of Tarski-Class X st
( v in L . C &amp; u c= v )  }  \/  {  (bool v) where v is Element of Tarski-Class X : v in L . C  }  ) \/ ((bool (L . C)) /\ (Tarski-Class X)) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = (union (rng (L | C))) /\ (Tarski-Class X) ) ) holds
b1 = b2;
proofend;
end;

:: deftheorem Def5 defines Tarski-Class CLASSES1:def_5_:_
 for X being set
for A being Ordinal
for b3 being set holds
( b3 = Tarski-Class (X,A) iff ex L being T-Sequence st
( b3 = last L &amp; dom L = succ A &amp; L . {} = {X} &amp; ( for C being Ordinal st succ C in succ A holds
L . (succ C) = ( {  u where u is Element of Tarski-Class X : ex v being Element of Tarski-Class X st
( v in L . C &amp; u c= v )  }  \/  {  (bool v) where v is Element of Tarski-Class X : v in L . C  }  ) \/ ((bool (L . C)) /\ (Tarski-Class X)) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = (union (rng (L | C))) /\ (Tarski-Class X) ) ) );

Lm1: now__::_thesis:_for_X_being_set_holds_
(_H1(_{}_)_=_{X}_&amp;_(_for_A_being_Ordinal_holds_H1(_succ_A)_=_H2(A,H1(A))_)_&amp;_(_for_A_being_Ordinal
for_L_being_T-Sequence_st_A_<>_{}_&amp;_A_is_limit_ordinal_&amp;_dom_L_=_A_&amp;_(_for_B_being_Ordinal_st_B_in_A_holds_
L_._B_=_Tarski-Class_(X,B)_)_holds_
Tarski-Class_(X,A)_=_(union_(rng_L))_/\_(Tarski-Class_X)_)_)
let X be set ; ::_thesis: ( H1( {} ) = {X} &amp; ( for A being Ordinal holds H1( succ A) = H2(A,H1(A)) ) &amp; ( for A being Ordinal
for L being T-Sequence st A <> {} &amp; A is limit_ordinal &amp; dom L = A &amp; ( for B being Ordinal st B in A holds
L . B = Tarski-Class (X,B) ) holds
Tarski-Class (X,A) = (union (rng L)) /\ (Tarski-Class X) ) )
deffunc H1( Ordinal) ->  set = Tarski-Class (X,$1);
deffunc H2( Ordinal, set ) ->  set = ( {  u where u is Element of Tarski-Class X : ex v being Element of Tarski-Class X st
( v in $2 &amp; u c= v )  }  \/  {  (bool v) where v is Element of Tarski-Class X : v in $2  }  ) \/ ((bool $2) /\ (Tarski-Class X));
deffunc H3( Ordinal, T-Sequence) ->  set = (union (rng $2)) /\ (Tarski-Class X);
A1: for A being Ordinal
for x being set holds
( x = H1(A) iff ex L being T-Sequence st
( x = last L &amp; dom L = succ A &amp; L . {} = {X} &amp; ( for C being Ordinal st succ C in succ A holds
L . (succ C) = H2(C,L . C) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = H3(C,L | C) ) ) ) by Def5;
thus  H1( {} ) = {X} from ORDINAL2:sch_8(A1); ::_thesis: ( ( for A being Ordinal holds H1( succ A) = H2(A,H1(A)) ) &amp; ( for A being Ordinal
for L being T-Sequence st A <> {} &amp; A is limit_ordinal &amp; dom L = A &amp; ( for B being Ordinal st B in A holds
L . B = Tarski-Class (X,B) ) holds
Tarski-Class (X,A) = (union (rng L)) /\ (Tarski-Class X) ) )
thus  for A being Ordinal holds H1( succ A) = H2(A,H1(A)) from ORDINAL2:sch_9(A1); ::_thesis: for A being Ordinal
for L being T-Sequence st A <> {} &amp; A is limit_ordinal &amp; dom L = A &amp; ( for B being Ordinal st B in A holds
L . B = Tarski-Class (X,B) ) holds
Tarski-Class (X,A) = (union (rng L)) /\ (Tarski-Class X)
thus  for A being Ordinal
for L being T-Sequence st A <> {} &amp; A is limit_ordinal &amp; dom L = A &amp; ( for B being Ordinal st B in A holds
L . B = Tarski-Class (X,B) ) holds
Tarski-Class (X,A) = (union (rng L)) /\ (Tarski-Class X) ::_thesis: verum
proof
let A be Ordinal; ::_thesis: for L being T-Sequence st A <> {} &amp; A is limit_ordinal &amp; dom L = A &amp; ( for B being Ordinal st B in A holds
L . B = Tarski-Class (X,B) ) holds
Tarski-Class (X,A) = (union (rng L)) /\ (Tarski-Class X)
let L be T-Sequence; ::_thesis: ( A <> {} &amp; A is limit_ordinal &amp; dom L = A &amp; ( for B being Ordinal st B in A holds
L . B = Tarski-Class (X,B) ) implies Tarski-Class (X,A) = (union (rng L)) /\ (Tarski-Class X) )
assume that
A2: ( A <> {} &amp; A is limit_ordinal ) and
A3: dom L = A and
A4: for B being Ordinal st B in A holds
L . B = H1(B) ; ::_thesis: Tarski-Class (X,A) = (union (rng L)) /\ (Tarski-Class X)
thus  H1(A) = H3(A,L) from ORDINAL2:sch_10(A1, A2, A3, A4); ::_thesis: verum
end;
end;

definition
let X be set ;
let A be Ordinal;
:: original:Tarski-Class
redefine func Tarski-Class (X,A) ->  Subset of (Tarski-Class X);
coherence
 Tarski-Class (X,A) is Subset of (Tarski-Class X)
proofend;
end;

theorem :: CLASSES1:7
for X being set holds Tarski-Class (X,{}) = {X} by Lm1;

theorem :: CLASSES1:8
for X being set
for A being Ordinal holds Tarski-Class (X,(succ A)) = ( {  u where u is Element of Tarski-Class X : ex v being Element of Tarski-Class X st
( v in Tarski-Class (X,A) &amp; u c= v )  }  \/  {  (bool v) where v is Element of Tarski-Class X : v in Tarski-Class (X,A)  }  ) \/ ((bool (Tarski-Class (X,A))) /\ (Tarski-Class X)) by Lm1;

theorem Th9: :: CLASSES1:9
for X being set
for A being Ordinal st A <> {} &amp; A is limit_ordinal holds
Tarski-Class (X,A) =  {  u where u is Element of Tarski-Class X : ex B being Ordinal st
( B in A &amp; u in Tarski-Class (X,B) )  }
proofend;

theorem Th10: :: CLASSES1:10
for Y, X being set
for A being Ordinal holds
( Y in Tarski-Class (X,(succ A)) iff ( ( Y c= Tarski-Class (X,A) &amp; Y in Tarski-Class X ) or ex Z being set st
( Z in Tarski-Class (X,A) &amp; ( Y c= Z or Y = bool Z ) ) ) )
proofend;

theorem :: CLASSES1:11
for Y, Z, X being set
for A being Ordinal st Y c= Z &amp; Z in Tarski-Class (X,A) holds
Y in Tarski-Class (X,(succ A)) by Th10;

theorem :: CLASSES1:12
for Y, X being set
for A being Ordinal st Y in Tarski-Class (X,A) holds
bool Y in Tarski-Class (X,(succ A)) by Th10;

theorem Th13: :: CLASSES1:13
for X, x being set
for A being Ordinal st A <> {} &amp; A is limit_ordinal holds
( x in Tarski-Class (X,A) iff ex B being Ordinal st
( B in A &amp; x in Tarski-Class (X,B) ) )
proofend;

theorem :: CLASSES1:14
for Y, X, Z being set
for A being Ordinal st A <> {} &amp; A is limit_ordinal &amp; Y in Tarski-Class (X,A) &amp; ( Z c= Y or Z = bool Y ) holds
Z in Tarski-Class (X,A)
proofend;

theorem Th15: :: CLASSES1:15
for X being set
for A being Ordinal holds Tarski-Class (X,A) c= Tarski-Class (X,(succ A))
proofend;

theorem Th16: :: CLASSES1:16
for X being set
for A, B being Ordinal st A c= B holds
Tarski-Class (X,A) c= Tarski-Class (X,B)
proofend;

theorem Th17: :: CLASSES1:17
for X being set ex A being Ordinal st Tarski-Class (X,A) = Tarski-Class (X,(succ A))
proofend;

theorem Th18: :: CLASSES1:18
for X being set
for A being Ordinal st Tarski-Class (X,A) = Tarski-Class (X,(succ A)) holds
Tarski-Class (X,A) = Tarski-Class X
proofend;

theorem Th19: :: CLASSES1:19
for X being set ex A being Ordinal st Tarski-Class (X,A) = Tarski-Class X
proofend;

theorem :: CLASSES1:20
for X being set ex A being Ordinal st
( Tarski-Class (X,A) = Tarski-Class X &amp; ( for B being Ordinal st B in A holds
Tarski-Class (X,B) <> Tarski-Class X ) )
proofend;

theorem :: CLASSES1:21
for Y, X being set st Y <> X &amp; Y in Tarski-Class X holds
ex A being Ordinal st
( not Y in Tarski-Class (X,A) &amp; Y in Tarski-Class (X,(succ A)) )
proofend;

theorem Th22: :: CLASSES1:22
for X being set st X is epsilon-transitive holds
for A being Ordinal st A <> {} holds
Tarski-Class (X,A) is epsilon-transitive
proofend;

theorem Th23: :: CLASSES1:23
for X being set st X is epsilon-transitive holds
Tarski-Class X is epsilon-transitive
proofend;

theorem Th24: :: CLASSES1:24
for Y, X being set st Y in Tarski-Class X holds
card Y in card (Tarski-Class X)
proofend;

theorem Th25: :: CLASSES1:25
for Y, X being set st Y in Tarski-Class X holds
not Y, Tarski-Class X are_equipotent
proofend;

theorem Th26: :: CLASSES1:26
for X, x, y being set st x in Tarski-Class X &amp; y in Tarski-Class X holds
( {x} in Tarski-Class X &amp; {x,y} in Tarski-Class X )
proofend;

theorem Th27: :: CLASSES1:27
for X, x, y being set st x in Tarski-Class X &amp; y in Tarski-Class X holds
[x,y] in Tarski-Class X
proofend;

theorem :: CLASSES1:28
for Y, X, Z being set st Y c= Tarski-Class X &amp; Z c= Tarski-Class X holds
[:Y,Z:] c= Tarski-Class X
proofend;

definition
let A be Ordinal;
func Rank A ->  set  means :Def6: :: CLASSES1:def 6
 ex L being T-Sequence st
( it = last L &amp; dom L = succ A &amp; L . {} = {} &amp; ( for C being Ordinal st succ C in succ A holds
L . (succ C) = bool (L . C) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = union (rng (L | C)) ) );
correctness
existence
ex b1 being set ex L being T-Sequence st
( b1 = last L &amp; dom L = succ A &amp; L . {} = {} &amp; ( for C being Ordinal st succ C in succ A holds
L . (succ C) = bool (L . C) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = union (rng (L | C)) ) );
uniqueness
for b1, b2 being set st ex L being T-Sequence st
( b1 = last L &amp; dom L = succ A &amp; L . {} = {} &amp; ( for C being Ordinal st succ C in succ A holds
L . (succ C) = bool (L . C) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = union (rng (L | C)) ) ) &amp; ex L being T-Sequence st
( b2 = last L &amp; dom L = succ A &amp; L . {} = {} &amp; ( for C being Ordinal st succ C in succ A holds
L . (succ C) = bool (L . C) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = union (rng (L | C)) ) ) holds
b1 = b2;
proofend;
end;

:: deftheorem Def6 defines Rank CLASSES1:def_6_:_
 for A being Ordinal
for b2 being set holds
( b2 = Rank A iff ex L being T-Sequence st
( b2 = last L &amp; dom L = succ A &amp; L . {} = {} &amp; ( for C being Ordinal st succ C in succ A holds
L . (succ C) = bool (L . C) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = union (rng (L | C)) ) ) );

deffunc H1( Ordinal) ->  set = Rank $1;

Lm2: now__::_thesis:_(_H1(_{}_)_=_{}_&amp;_(_for_A_being_Ordinal_holds_H1(_succ_A)_=_H2(A,H1(A))_)_&amp;_(_for_A_being_Ordinal
for_L_being_T-Sequence_st_A_<>_{}_&amp;_A_is_limit_ordinal_&amp;_dom_L_=_A_&amp;_(_for_B_being_Ordinal_st_B_in_A_holds_
L_._B_=_Rank_B_)_holds_
Rank_A_=_union_(rng_L)_)_)
deffunc H2( Ordinal, set ) ->  Element of bool (bool $2) = bool $2;
deffunc H3( Ordinal, T-Sequence) ->  set = union (rng $2);
A1: for A being Ordinal
for x being set holds
( x = H1(A) iff ex L being T-Sequence st
( x = last L &amp; dom L = succ A &amp; L . {} = {} &amp; ( for C being Ordinal st succ C in succ A holds
L . (succ C) = H2(C,L . C) ) &amp; ( for C being Ordinal st C in succ A &amp; C <> {} &amp; C is limit_ordinal holds
L . C = H3(C,L | C) ) ) ) by Def6;
thus  H1( {} ) = {} from ORDINAL2:sch_8(A1); ::_thesis: ( ( for A being Ordinal holds H1( succ A) = H2(A,H1(A)) ) &amp; ( for A being Ordinal
for L being T-Sequence st A <> {} &amp; A is limit_ordinal &amp; dom L = A &amp; ( for B being Ordinal st B in A holds
L . B = Rank B ) holds
Rank A = union (rng L) ) )
thus  for A being Ordinal holds H1( succ A) = H2(A,H1(A)) from ORDINAL2:sch_9(A1); ::_thesis: for A being Ordinal
for L being T-Sequence st A <> {} &amp; A is limit_ordinal &amp; dom L = A &amp; ( for B being Ordinal st B in A holds
L . B = Rank B ) holds
Rank A = union (rng L)
thus  for A being Ordinal
for L being T-Sequence st A <> {} &amp; A is limit_ordinal &amp; dom L = A &amp; ( for B being Ordinal st B in A holds
L . B = Rank B ) holds
Rank A = union (rng L) ::_thesis: verum
proof
let A be Ordinal; ::_thesis: for L being T-Sequence st A <> {} &amp; A is limit_ordinal &amp; dom L = A &amp; ( for B being Ordinal st B in A holds
L . B = Rank B ) holds
Rank A = union (rng L)
let L be T-Sequence; ::_thesis: ( A <> {} &amp; A is limit_ordinal &amp; dom L = A &amp; ( for B being Ordinal st B in A holds
L . B = Rank B ) implies Rank A = union (rng L) )
assume that
A2: ( A <> {} &amp; A is limit_ordinal ) and
A3: dom L = A and
A4: for B being Ordinal st B in A holds
L . B = H1(B) ; ::_thesis: Rank A = union (rng L)
thus  H1(A) = H3(A,L) from ORDINAL2:sch_10(A1, A2, A3, A4); ::_thesis: verum
end;
end;

theorem :: CLASSES1:29
Rank {} = {} by Lm2;

theorem :: CLASSES1:30
for A being Ordinal holds Rank (succ A) = bool (Rank A) by Lm2;

theorem Th31: :: CLASSES1:31
for A being Ordinal st A <> {} &amp; A is limit_ordinal holds
for x being set holds
( x in Rank A iff ex B being Ordinal st
( B in A &amp; x in Rank B ) )
proofend;

theorem Th32: :: CLASSES1:32
for X being set
for A being Ordinal holds
( X c= Rank A iff X in Rank (succ A) )
proofend;

registration
let A be Ordinal;
cluster Rank A ->  epsilon-transitive ;
coherence
 Rank A is epsilon-transitive
proofend;
end;

theorem :: CLASSES1:33
for A being Ordinal holds Rank A c= Rank (succ A)
proofend;

theorem Th34: :: CLASSES1:34
for A being Ordinal holds union (Rank A) c= Rank A
proofend;

theorem :: CLASSES1:35
for X being set
for A being Ordinal st X in Rank A holds
union X in Rank A
proofend;

theorem Th36: :: CLASSES1:36
for A, B being Ordinal holds
( A in B iff Rank A in Rank B )
proofend;

theorem Th37: :: CLASSES1:37
for A, B being Ordinal holds
( A c= B iff Rank A c= Rank B )
proofend;

theorem Th38: :: CLASSES1:38
for A being Ordinal holds A c= Rank A
proofend;

theorem :: CLASSES1:39
for A being Ordinal
for X being set st X in Rank A holds
( not X, Rank A are_equipotent &amp; card X in card (Rank A) )
proofend;

theorem :: CLASSES1:40
for X being set
for A being Ordinal holds
( X c= Rank A iff bool X c= Rank (succ A) )
proofend;

theorem Th41: :: CLASSES1:41
for X, Y being set
for A being Ordinal st X c= Y &amp; Y in Rank A holds
X in Rank A
proofend;

theorem Th42: :: CLASSES1:42
for X being set
for A being Ordinal holds
( X in Rank A iff bool X in Rank (succ A) )
proofend;

theorem Th43: :: CLASSES1:43
for x being set
for A being Ordinal holds
( x in Rank A iff {x} in Rank (succ A) )
proofend;

theorem Th44: :: CLASSES1:44
for x, y being set
for A being Ordinal holds
( ( x in Rank A &amp; y in Rank A ) iff {x,y} in Rank (succ A) )
proofend;

theorem :: CLASSES1:45
for x, y being set
for A being Ordinal holds
( ( x in Rank A &amp; y in Rank A ) iff [x,y] in Rank (succ (succ A)) )
proofend;

theorem Th46: :: CLASSES1:46
for X being set
for A being Ordinal st X is epsilon-transitive &amp; (Rank A) /\ (Tarski-Class X) = (Rank (succ A)) /\ (Tarski-Class X) holds
Tarski-Class X c= Rank A
proofend;

theorem Th47: :: CLASSES1:47
for X being set st X is epsilon-transitive holds
ex A being Ordinal st Tarski-Class X c= Rank A
proofend;

theorem Th48: :: CLASSES1:48
for X being set st X is epsilon-transitive holds
union X c= X
proofend;

theorem Th49: :: CLASSES1:49
for X, Y being set st X is epsilon-transitive &amp; Y is epsilon-transitive holds
X \/ Y is epsilon-transitive
proofend;

theorem :: CLASSES1:50
for X, Y being set st X is epsilon-transitive &amp; Y is epsilon-transitive holds
X /\ Y is epsilon-transitive
proofend;

deffunc H2( set , set ) ->  set = union $2;

definition
let X be set ;
func the_transitive-closure_of X ->  set  means :Def7: :: CLASSES1:def 7
 for x being set holds
( x in it iff ex f being Function ex n being Element of omega st
( x in f . n &amp; dom f = omega &amp; f . 0 = X &amp; ( for k being Nat holds f . (succ k) = union (f . k) ) ) );
existence
 ex b1 being set st
for x being set holds
( x in b1 iff ex f being Function ex n being Element of omega st
( x in f . n &amp; dom f = omega &amp; f . 0 = X &amp; ( for k being Nat holds f . (succ k) = union (f . k) ) ) )
proofend;
uniqueness
 for b1, b2 being set st ( for x being set holds
( x in b1 iff ex f being Function ex n being Element of omega st
( x in f . n &amp; dom f = omega &amp; f . 0 = X &amp; ( for k being Nat holds f . (succ k) = union (f . k) ) ) ) ) &amp; ( for x being set holds
( x in b2 iff ex f being Function ex n being Element of omega st
( x in f . n &amp; dom f = omega &amp; f . 0 = X &amp; ( for k being Nat holds f . (succ k) = union (f . k) ) ) ) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def7 defines the_transitive-closure_of CLASSES1:def_7_:_
 for X, b2 being set holds
( b2 = the_transitive-closure_of X iff for x being set holds
( x in b2 iff ex f being Function ex n being Element of omega st
( x in f . n &amp; dom f = omega &amp; f . 0 = X &amp; ( for k being Nat holds f . (succ k) = union (f . k) ) ) ) );

theorem Th51: :: CLASSES1:51
for X being set holds the_transitive-closure_of X is epsilon-transitive
proofend;

theorem Th52: :: CLASSES1:52
for X being set holds X c= the_transitive-closure_of X
proofend;

theorem Th53: :: CLASSES1:53
for X, Y being set st X c= Y &amp; Y is epsilon-transitive holds
the_transitive-closure_of X c= Y
proofend;

theorem Th54: :: CLASSES1:54
for X, Y being set st ( for Z being set st X c= Z &amp; Z is epsilon-transitive holds
Y c= Z ) &amp; X c= Y &amp; Y is epsilon-transitive holds
the_transitive-closure_of X = Y
proofend;

theorem Th55: :: CLASSES1:55
for X being set st X is epsilon-transitive holds
the_transitive-closure_of X = X
proofend;

theorem :: CLASSES1:56
the_transitive-closure_of {} = {} by Th55;

theorem :: CLASSES1:57
for A being Ordinal holds the_transitive-closure_of A = A by Th55;

theorem Th58: :: CLASSES1:58
for X, Y being set st X c= Y holds
the_transitive-closure_of X c= the_transitive-closure_of Y
proofend;

theorem :: CLASSES1:59
for X being set holds the_transitive-closure_of (the_transitive-closure_of X) = the_transitive-closure_of X by Th51, Th55;

theorem :: CLASSES1:60
for X, Y being set holds the_transitive-closure_of (X \/ Y) = (the_transitive-closure_of X) \/ (the_transitive-closure_of Y)
proofend;

theorem :: CLASSES1:61
for X, Y being set holds the_transitive-closure_of (X /\ Y) c= (the_transitive-closure_of X) /\ (the_transitive-closure_of Y)
proofend;

theorem Th62: :: CLASSES1:62
for X being set ex A being Ordinal st X c= Rank A
proofend;

definition
let X be set ;
func the_rank_of X ->  Ordinal means :Def8: :: CLASSES1:def 8
( X c= Rank it &amp; ( for B being Ordinal st X c= Rank B holds
it c= B ) );
existence
 ex b1 being Ordinal st
( X c= Rank b1 &amp; ( for B being Ordinal st X c= Rank B holds
b1 c= B ) )
proofend;
uniqueness
 for b1, b2 being Ordinal st X c= Rank b1 &amp; ( for B being Ordinal st X c= Rank B holds
b1 c= B ) &amp; X c= Rank b2 &amp; ( for B being Ordinal st X c= Rank B holds
b2 c= B ) holds
b1 = b2
proofend;
end;

:: deftheorem Def8 defines the_rank_of CLASSES1:def_8_:_
 for X being set
for b2 being Ordinal holds
( b2 = the_rank_of X iff ( X c= Rank b2 &amp; ( for B being Ordinal st X c= Rank B holds
b2 c= B ) ) );

theorem Th63: :: CLASSES1:63
for X being set holds the_rank_of (bool X) = succ (the_rank_of X)
proofend;

theorem :: CLASSES1:64
for A being Ordinal holds the_rank_of (Rank A) = A
proofend;

theorem Th65: :: CLASSES1:65
for X being set
for A being Ordinal holds
( X c= Rank A iff the_rank_of X c= A )
proofend;

theorem Th66: :: CLASSES1:66
for X being set
for A being Ordinal holds
( X in Rank A iff the_rank_of X in A )
proofend;

theorem :: CLASSES1:67
for X, Y being set st X c= Y holds
the_rank_of X c= the_rank_of Y
proofend;

theorem Th68: :: CLASSES1:68
for X, Y being set st X in Y holds
the_rank_of X in the_rank_of Y
proofend;

theorem Th69: :: CLASSES1:69
for X being set
for A being Ordinal holds
( the_rank_of X c= A iff for Y being set st Y in X holds
the_rank_of Y in A )
proofend;

theorem Th70: :: CLASSES1:70
for X being set
for A being Ordinal holds
( A c= the_rank_of X iff for B being Ordinal st B in A holds
ex Y being set st
( Y in X &amp; B c= the_rank_of Y ) )
proofend;

theorem :: CLASSES1:71
for X being set holds
( the_rank_of X = {} iff X = {} )
proofend;

theorem Th72: :: CLASSES1:72
for X being set
for A being Ordinal st the_rank_of X = succ A holds
ex Y being set st
( Y in X &amp; the_rank_of Y = A )
proofend;

theorem :: CLASSES1:73
for A being Ordinal holds the_rank_of A = A
proofend;

theorem :: CLASSES1:74
for X being set holds
( the_rank_of (Tarski-Class X) <> {} &amp; the_rank_of (Tarski-Class X) is limit_ordinal )
proofend;

begin

scheme :: CLASSES1:sch 1
NonUniqFuncEx{ F1() ->  set , P1[ set , set ] } :
 ex f being Function st
( dom f = F1() &amp; ( for e being set st e in F1() holds
P1[e,f . e] ) )
provided
A1: for e being set st e in F1() holds
ex u being set st P1[e,u]
proofend;

:: from RFINSEQ, 2008.10.10, A.T.
definition
let F, G be Relation;
predF,G are_fiberwise_equipotent  means :Def9: :: CLASSES1:def 9
 for x being set holds card (Coim (F,x)) = card (Coim (G,x));
reflexivity
 for F being Relation
for x being set holds card (Coim (F,x)) = card (Coim (F,x)) ;
symmetry
 for F, G being Relation st ( for x being set holds card (Coim (F,x)) = card (Coim (G,x)) ) holds
for x being set holds card (Coim (G,x)) = card (Coim (F,x)) ;
end;

:: deftheorem Def9 defines are_fiberwise_equipotent CLASSES1:def_9_:_
 for F, G being Relation holds
( F,G are_fiberwise_equipotent iff for x being set holds card (Coim (F,x)) = card (Coim (G,x)) );

Lm3:  for F being Function
for x being set st not x in rng F holds
Coim (F,x) = {}
proofend;

theorem Th75: :: CLASSES1:75
for F, G being Function st F,G are_fiberwise_equipotent holds
rng F = rng G
proofend;

theorem :: CLASSES1:76
for F, G, H being Function st F,G are_fiberwise_equipotent &amp; F,H are_fiberwise_equipotent holds
G,H are_fiberwise_equipotent
proofend;

theorem Th77: :: CLASSES1:77
for F, G being Function holds
( F,G are_fiberwise_equipotent iff ex H being Function st
( dom H = dom F &amp; rng H = dom G &amp; H is one-to-one &amp; F = G * H ) )
proofend;

theorem Th78: :: CLASSES1:78
for F, G being Function holds
( F,G are_fiberwise_equipotent iff for X being set holds card (F " X) = card (G " X) )
proofend;

theorem :: CLASSES1:79
for D being non empty set
for F, G being Function st rng F c= D &amp; rng G c= D &amp; ( for d being Element of D holds card (Coim (F,d)) = card (Coim (G,d)) ) holds
F,G are_fiberwise_equipotent
proofend;

theorem Th80: :: CLASSES1:80
for F, G being Function st dom F = dom G holds
( F,G are_fiberwise_equipotent iff ex P being Permutation of (dom F) st F = G * P )
proofend;

theorem :: CLASSES1:81
for F, G being Function st F,G are_fiberwise_equipotent holds
card (dom F) = card (dom G)
proofend;

:: from CIRCCOMB, 2009.01.26, A.T.
theorem :: CLASSES1:82
for f being set
for p being Relation
for x being set st x in rng p holds
the_rank_of x in the_rank_of [p,f]
proofend;

:: from BAGORDER, 2011.07.24, A.T.
theorem :: CLASSES1:83
for f, g, h being Function st dom f = dom g &amp; rng f c= dom h &amp; rng g c= dom h &amp; f,g are_fiberwise_equipotent holds
h * f,h * g are_fiberwise_equipotent
proofend;