:: CLASSES2 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

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

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

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

theorem Th1: :: CLASSES2:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W, X being set st W is subset-closed & X in W holds
( not X,W are_equipotent & Card X <` Card W )
proof end;

theorem :: CLASSES2:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th3: :: CLASSES2:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W, x, y being set st W is_Tarski-Class & x in W & y in W holds
( {x} in W & {x,y} in W )
proof end;

theorem Th4: :: CLASSES2:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W, x, y being set st W is_Tarski-Class & x in W & y in W holds
[x,y] in W
proof end;

theorem Th5: :: CLASSES2:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W, X being set st W is_Tarski-Class & X in W holds
Tarski-Class X c= W
proof end;

theorem Th6: :: CLASSES2:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being Ordinal
for W being set st W is_Tarski-Class & A in W holds
( succ A in W & A c= W )
proof end;

theorem :: CLASSES2:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 Th6;

theorem Th8: :: CLASSES2:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W, X being set st W is subset-closed & X is epsilon-transitive & X in W holds
X c= W
proof end;

theorem :: CLASSES2:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X, W being set st X is epsilon-transitive & X in Tarski-Class W holds
X c= Tarski-Class W by Th8;

theorem Th10: :: CLASSES2:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set st W is_Tarski-Class holds
On W = Card W
proof end;

theorem :: CLASSES2:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set holds On (Tarski-Class W) = Card (Tarski-Class W) by Th10;

theorem Th12: :: CLASSES2:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W, X being set st W is_Tarski-Class & X in W holds
Card X in W
proof end;

theorem :: CLASSES2:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X, W being set st X in Tarski-Class W holds
Card X in Tarski-Class W by Th12;

theorem Th14: :: CLASSES2:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W, x being set st W is_Tarski-Class & x in Card W holds
x in W
proof end;

theorem :: CLASSES2:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for x, W being set st x in Card (Tarski-Class W) holds
x in Tarski-Class W by Th14;

theorem :: CLASSES2:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for m being Cardinal
for W being set st W is_Tarski-Class & m <` Card W holds
m in W
proof end;

theorem :: CLASSES2:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for m being Cardinal
for W being set st m <` Card (Tarski-Class W) holds
m in Tarski-Class W
proof end;

theorem :: CLASSES2:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for m being Cardinal
for W being set st W is_Tarski-Class & m in W holds
m c= W by Th6;

theorem :: CLASSES2:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for m being Cardinal
for W being set st m in Tarski-Class W holds
m c= Tarski-Class W by Th6;

theorem Th20: :: CLASSES2:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set st W is_Tarski-Class holds
Card W is_limit_ordinal
proof end;

theorem Th21: :: CLASSES2:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set st W is_Tarski-Class & W <> {} holds
( Card W <> 0 & Card W <> {} & Card W is_limit_ordinal )
proof end;

theorem Th22: :: CLASSES2:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set holds
( Card (Tarski-Class W) <> 0 & Card (Tarski-Class W) <> {} & Card (Tarski-Class W) is_limit_ordinal )
proof end;

theorem Th23: :: CLASSES2:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W, X being set st W is_Tarski-Class & ( ( X in W & W is epsilon-transitive ) or ( X in W & X c= W ) or ( Card X <` Card W & X c= W ) ) holds
Funcs X,W c= W
proof end;

theorem :: CLASSES2:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 <` Card (Tarski-Class W) & X c= Tarski-Class W ) ) holds
Funcs X,(Tarski-Class W) c= Tarski-Class W
proof end;

theorem Th25: :: CLASSES2:25  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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-Class & $1 in On W holds
( Card (Rank $1) <` Card W & Rank $1 in W );

Lm2: S1[ {} ]
by Th10, CARD_1:47, CLASSES1:33, ORDINAL2:def 2;

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 Th26: :: CLASSES2:26  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being Ordinal
for W being set st W is_Tarski-Class & A in On W holds
( Card (Rank A) <` Card W & Rank A in W )
proof end;

theorem :: CLASSES2:27  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being Ordinal
for W being set st A in On (Tarski-Class W) holds
( Card (Rank A) <` Card (Tarski-Class W) & Rank A in Tarski-Class W ) by Th26;

theorem Th28: :: CLASSES2:28  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set st W is_Tarski-Class holds
Rank (Card W) c= W
proof end;

theorem Th29: :: CLASSES2:29  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 Th30: :: CLASSES2:30  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W, X being set st W is_Tarski-Class & W is epsilon-transitive & X in W holds
the_rank_of X in W
proof end;

theorem Th31: :: CLASSES2:31  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set st W is_Tarski-Class & W is epsilon-transitive holds
W c= Rank (Card W)
proof end;

theorem Th32: :: CLASSES2:32  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set st W is_Tarski-Class & W is epsilon-transitive holds
Rank (Card W) = W
proof end;

theorem :: CLASSES2:33  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being Ordinal
for W being set st W is_Tarski-Class & A in On W holds
Card (Rank A) <=` Card W
proof end;

theorem :: CLASSES2:34  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being Ordinal
for W being set st A in On (Tarski-Class W) holds
Card (Rank A) <=` Card (Tarski-Class W)
proof end;

theorem Th35: :: CLASSES2:35  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set st W is_Tarski-Class holds
Card W = Card (Rank (Card W))
proof end;

theorem :: CLASSES2:36  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set holds Card (Tarski-Class W) = Card (Rank (Card (Tarski-Class W))) by Th35;

theorem Th37: :: CLASSES2:37  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W, X being set st W is_Tarski-Class & X c= Rank (Card W) & not X, Rank (Card W) are_equipotent holds
X in Rank (Card W)
proof end;

theorem :: CLASSES2:38  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 Th37;

theorem Th39: :: CLASSES2:39  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set st W is_Tarski-Class holds
Rank (Card W) is_Tarski-Class
proof end;

theorem :: CLASSES2:40  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set holds Rank (Card (Tarski-Class W)) is_Tarski-Class by Th39;

theorem Th41: :: CLASSES2:41  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 Th42: :: CLASSES2:42  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being set st X is epsilon-transitive holds
Card (the_rank_of X) <=` Card X
proof end;

theorem Th43: :: CLASSES2:43  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W, X being set st W is_Tarski-Class & X is epsilon-transitive & X in W holds
X in Rank (Card W)
proof end;

theorem :: CLASSES2:44  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X, W being set st X is epsilon-transitive & X in Tarski-Class W holds
X in Rank (Card (Tarski-Class W)) by Th43;

theorem Th45: :: CLASSES2:45  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for W being set st W is epsilon-transitive holds
Rank (Card (Tarski-Class W)) is_Tarski-Class_of W
proof end;

theorem :: CLASSES2:46  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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-Class );
end;

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

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

registration
cluster epsilon-transitive non empty subset-closed being_Tarski-Class universal 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 :: CLASSES2:47  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: CLASSES2:48  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: CLASSES2:49  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th50: :: CLASSES2:50  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for U being Universe holds On U is Ordinal
proof end;

theorem Th51: :: CLASSES2:51  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for X being set st X is epsilon-transitive holds
Tarski-Class X is universal
proof end;

theorem :: CLASSES2:52  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for U being Universe holds Tarski-Class U is Universe by Th51;

registration
let U be Universe;
cluster On U -> ordinal ;
coherence
On U is ordinal
by Th50;
cluster Tarski-Class U -> epsilon-transitive subset-closed being_Tarski-Class universal ;
coherence
Tarski-Class U is universal
by Th51;
end;

theorem Th53: :: CLASSES2:53  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being Ordinal holds Tarski-Class A is universal
proof end;

registration
let A be Ordinal;
cluster Tarski-Class A -> epsilon-transitive subset-closed being_Tarski-Class universal ;
coherence
Tarski-Class A is universal
by Th53;
end;

theorem Th54: :: CLASSES2:54  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for U being Universe holds U = Rank (On U)
proof end;

theorem Th55: :: CLASSES2:55  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for U being Universe holds
( On U <> {} & On U is_limit_ordinal )
proof end;

theorem :: CLASSES2:56  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for U1, U2 being Universe holds
( U1 in U2 or U1 = U2 or U2 in U1 )
proof end;

theorem Th57: :: CLASSES2:57  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for U1, U2 being Universe holds
( U1 c= U2 or U2 in U1 )
proof end;

theorem Th58: :: CLASSES2:58  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for U1, U2 being Universe holds U1,U2 are_c=-comparable
proof end;

theorem Th59: :: CLASSES2:59  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for U1, U2, U3 being Universe st U1 in U2 & U2 in U3 holds
U1 in U3
proof end;

theorem :: CLASSES2:60  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: CLASSES2:61  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for U1, U2 being Universe holds
( U1 \/ U2 is Universe & U1 /\ U2 is Universe )
proof end;

theorem Th62: :: CLASSES2:62  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for U being Universe holds {} in U
proof end;

theorem :: CLASSES2:63  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for x being set
for U being Universe st x in U holds
{x} in U by Th3;

theorem :: CLASSES2:64  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 Th3, Th4;

theorem Th65: :: CLASSES2:65  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 Th66: :: CLASSES2:66  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 Th67: :: CLASSES2:67  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 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 Th3;
:: original: bool
redefine func bool u -> non empty Element of U;
coherence
bool u is non empty Element of U
by Th65;
:: original: union
redefine func union u -> Element of U;
coherence
union u is Element of U
by Th65;
:: original: meet
redefine func meet u -> Element of U;
coherence
meet u is Element of U
by Th65;
let v be Element of U;
:: 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 Th4;
:: original: \/
redefine func u \/ v -> Element of U;
coherence
u \/ v is Element of U
by Th66;
:: original: /\
redefine func u /\ v -> Element of U;
coherence
u /\ v is Element of U
by Th66;
:: original: \
redefine func u \ v -> Element of U;
coherence
u \ v is Element of U
by Th66;
:: original: \+\
redefine func u \+\ v -> Element of U;
coherence
u \+\ v is Element of U
by Th66;
:: original: [:
redefine func [:u,v:] -> Element of U;
coherence
[:u,v:] is Element of U
by Th67;
:: original: Funcs
redefine func Funcs u,v -> Element of U;
coherence
Funcs u,v is Element of U
by Th67;
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 {} ;

theorem :: CLASSES2:68  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th69: :: CLASSES2:69  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
Card (Rank omega ) = Card omega
proof end;

theorem Th70: :: CLASSES2:70  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
Rank omega is_Tarski-Class
proof end;

theorem :: CLASSES2:71  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 CLASSES1:58;
end;

registration
let X be epsilon-transitive set ;
cluster Tarski-Class X -> epsilon-transitive subset-closed being_Tarski-Class universal ;
coherence
Tarski-Class X is epsilon-transitive
by CLASSES1:27;
end;

registration
let A be Ordinal;
cluster Rank A -> epsilon-transitive ;
coherence
Rank A is epsilon-transitive
by CLASSES1:37;
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;

Lm5: now
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( FinSETS , 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( FinSETS , 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( B A FinSETS , A1, A2, A3, A4); :: thesis: verum
end;

registration
let A be Ordinal;
cluster UNIVERSE A -> epsilon-transitive non empty subset-closed being_Tarski-Class universal ;
coherence
( UNIVERSE A is universal & not UNIVERSE A is empty )
proof end;
end;

theorem :: CLASSES2:72  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: CLASSES2:73  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: CLASSES2:74  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: CLASSES2:75  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
UNIVERSE {} = FinSETS by Lm5;

theorem :: CLASSES2:76  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being Ordinal holds UNIVERSE (succ A) = Tarski-Class (UNIVERSE A) by Lm5;

theorem :: CLASSES2:77  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
UNIVERSE one = SETS by Lm5, ORDINAL2:def 4;

theorem :: CLASSES2:78  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 Lm5;

theorem :: CLASSES2:79  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for U being Universe holds
( FinSETS c= U & Tarski-Class {} c= U & UNIVERSE {} c= U )
proof end;

theorem Th80: :: CLASSES2:80  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A, B being Ordinal holds
( A in B iff UNIVERSE A in UNIVERSE B )
proof end;

theorem :: CLASSES2:81  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A, B being Ordinal st UNIVERSE A = UNIVERSE B holds
A = B
proof end;

theorem :: CLASSES2:82  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A, B being Ordinal holds
( A c= B iff UNIVERSE A c= UNIVERSE B )
proof end;