:: ZFREFLE1 semantic presentation

Lemma1: 1 = succ 0
;

definition
let c1 be non empty set ;
let c2 be Subset of WFF ;
pred c1 |= c2 means :Def1: :: ZFREFLE1:def 1
for b1 being ZF-formula holds
( b1 in a2 implies a1 |= b1 );
end;

:: deftheorem Def1 defines |= ZFREFLE1:def 1 :
for b1 being non empty set
for b2 being Subset of WFF holds
( b1 |= b2 iff for b3 being ZF-formula holds
( b3 in b2 implies b1 |= b3 ) );

definition
let c1, c2 be non empty set ;
pred c1 <==> c2 means :: ZFREFLE1:def 2
for b1 being ZF-formula holds
( Free b1 = {} implies ( a1 |= b1 iff a2 |= b1 ) );
reflexivity
for b1 being non empty set
for b2 being ZF-formula holds
( Free b2 = {} implies ( b1 |= b2 iff b1 |= b2 ) )
;
symmetry
for b1, b2 being non empty set holds
( ( for b3 being ZF-formula holds
( Free b3 = {} implies ( b1 |= b3 iff b2 |= b3 ) ) ) implies for b3 being ZF-formula holds
( Free b3 = {} implies ( b2 |= b3 iff b1 |= b3 ) ) )
;
pred c1 is_elementary_subsystem_of c2 means :: ZFREFLE1:def 3
( a1 c= a2 & ( for b1 being ZF-formula
for b2 being Function of VAR ,a1 holds
( a1,b2 |= b1 iff a2,a2 ! b2 |= b1 ) ) );
reflexivity
for b1 being non empty set holds
( b1 c= b1 & ( for b2 being ZF-formula
for b3 being Function of VAR ,b1 holds
( b1,b3 |= b2 iff b1,b1 ! b3 |= b2 ) ) )
by ZF_LANG1:def 2;
end;

:: deftheorem Def2 defines <==> ZFREFLE1:def 2 :
for b1, b2 being non empty set holds
( b1 <==> b2 iff for b3 being ZF-formula holds
( Free b3 = {} implies ( b1 |= b3 iff b2 |= b3 ) ) );

:: deftheorem Def3 defines is_elementary_subsystem_of ZFREFLE1:def 3 :
for b1, b2 being non empty set holds
( b1 is_elementary_subsystem_of b2 iff ( b1 c= b2 & ( for b3 being ZF-formula
for b4 being Function of VAR ,b1 holds
( b1,b4 |= b3 iff b2,b2 ! b4 |= b3 ) ) ) );

defpred S1[ set ] means not ( not a1 = the_axiom_of_extensionality & not a1 = the_axiom_of_pairs & not a1 = the_axiom_of_unions & not a1 = the_axiom_of_infinity & not a1 = the_axiom_of_power_sets & ( for b1 being ZF-formula holds
not ( {(x. 0),(x. 1),(x. 2)} misses Free b1 & a1 = the_axiom_of_substitution_for b1 ) ) );

definition
func ZF-axioms -> set means :Def4: :: ZFREFLE1:def 4
for b1 being set holds
( b1 in a1 iff ( b1 in WFF & not ( not b1 = the_axiom_of_extensionality & not b1 = the_axiom_of_pairs & not b1 = the_axiom_of_unions & not b1 = the_axiom_of_infinity & not b1 = the_axiom_of_power_sets & ( for b2 being ZF-formula holds
not ( {(x. 0),(x. 1),(x. 2)} misses Free b2 & b1 = the_axiom_of_substitution_for b2 ) ) ) ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ( b2 in WFF & not ( not b2 = the_axiom_of_extensionality & not b2 = the_axiom_of_pairs & not b2 = the_axiom_of_unions & not b2 = the_axiom_of_infinity & not b2 = the_axiom_of_power_sets & ( for b3 being ZF-formula holds
not ( {(x. 0),(x. 1),(x. 2)} misses Free b3 & b2 = the_axiom_of_substitution_for b3 ) ) ) ) )
proof end;
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ( b3 in WFF & not ( not b3 = the_axiom_of_extensionality & not b3 = the_axiom_of_pairs & not b3 = the_axiom_of_unions & not b3 = the_axiom_of_infinity & not b3 = the_axiom_of_power_sets & ( for b4 being ZF-formula holds
not ( {(x. 0),(x. 1),(x. 2)} misses Free b4 & b3 = the_axiom_of_substitution_for b4 ) ) ) ) ) ) & ( for b3 being set holds
( b3 in b2 iff ( b3 in WFF & not ( not b3 = the_axiom_of_extensionality & not b3 = the_axiom_of_pairs & not b3 = the_axiom_of_unions & not b3 = the_axiom_of_infinity & not b3 = the_axiom_of_power_sets & ( for b4 being ZF-formula holds
not ( {(x. 0),(x. 1),(x. 2)} misses Free b4 & b3 = the_axiom_of_substitution_for b4 ) ) ) ) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def4 defines ZF-axioms ZFREFLE1:def 4 :
for b1 being set holds
( b1 = ZF-axioms iff for b2 being set holds
( b2 in b1 iff ( b2 in WFF & not ( not b2 = the_axiom_of_extensionality & not b2 = the_axiom_of_pairs & not b2 = the_axiom_of_unions & not b2 = the_axiom_of_infinity & not b2 = the_axiom_of_power_sets & ( for b3 being ZF-formula holds
not ( {(x. 0),(x. 1),(x. 2)} misses Free b3 & b2 = the_axiom_of_substitution_for b3 ) ) ) ) ) );

definition
redefine func ZF-axioms as ZF-axioms -> Subset of WFF ;
coherence
ZF-axioms is Subset of WFF
proof end;
end;

theorem Th1: :: ZFREFLE1:1
for b1 being non empty set holds b1 |= {} WFF
proof end;

theorem Th2: :: ZFREFLE1:2
for b1 being non empty set
for b2, b3 being Subset of WFF holds
( b2 c= b3 & b1 |= b3 implies b1 |= b2 )
proof end;

theorem Th3: :: ZFREFLE1:3
for b1 being non empty set
for b2, b3 being Subset of WFF holds
( b1 |= b2 & b1 |= b3 implies b1 |= b2 \/ b3 )
proof end;

theorem Th4: :: ZFREFLE1:4
for b1 being non empty set holds
( b1 is_a_model_of_ZF implies b1 |= ZF-axioms )
proof end;

theorem Th5: :: ZFREFLE1:5
for b1 being non empty set holds
( b1 |= ZF-axioms & b1 is epsilon-transitive implies b1 is_a_model_of_ZF )
proof end;

theorem Th6: :: ZFREFLE1:6
for b1 being ZF-formula holds
ex b2 being ZF-formula st
( Free b2 = {} & ( for b3 being non empty set holds
( b3 |= b2 iff b3 |= b1 ) ) )
proof end;

theorem Th7: :: ZFREFLE1:7
for b1, b2 being non empty set holds
( b1 <==> b2 iff for b3 being ZF-formula holds
( b1 |= b3 iff b2 |= b3 ) )
proof end;

theorem Th8: :: ZFREFLE1:8
for b1, b2 being non empty set holds
( b1 <==> b2 iff for b3 being Subset of WFF holds
( b1 |= b3 iff b2 |= b3 ) )
proof end;

theorem Th9: :: ZFREFLE1:9
for b1, b2 being non empty set holds
( b1 is_elementary_subsystem_of b2 implies b1 <==> b2 )
proof end;

theorem Th10: :: ZFREFLE1:10
for b1, b2 being non empty set holds
( b1 is_a_model_of_ZF & b1 <==> b2 & b2 is epsilon-transitive implies b2 is_a_model_of_ZF )
proof end;

scheme :: ZFREFLE1:sch 1
s1{ F1() -> set , P1[ set , set ] } :
ex b1 being Function st
( dom b1 = F1() & ( for b2 being set holds
( b2 in F1() implies P1[b2,b1 . b2] ) ) )
provided
E10: for b1 being set holds
not ( b1 in F1() & ( for b2 being set holds
not P1[b1,b2] ) )
proof end;

theorem Th11: :: ZFREFLE1:11
canceled;

theorem Th12: :: ZFREFLE1:12
for b1 being Function
for b2 being Universe holds
( dom b1 in b2 & rng b1 c= b2 implies rng b1 in b2 )
proof end;

theorem Th13: :: ZFREFLE1:13
for b1, b2 being set holds
( ( b1,b2 are_equipotent or Card b1 = Card b2 ) implies ( bool b1, bool b2 are_equipotent & Card (bool b1) = Card (bool b2) ) )
proof end;

theorem Th14: :: ZFREFLE1:14
for b1 being Universe
for b2 being non empty set
for b3 being Function of b2, Funcs (On b1),(On b1) holds
not ( Card b2 <` Card b1 & ( for b4 being Ordinal-Sequence of b1 holds
not ( b4 is increasing & b4 is continuous & b4 . (0-element_of b1) = 0-element_of b1 & ( for b5 being Ordinal of b1 holds b4 . (succ b5) = sup ({(b4 . b5)} \/ ((uncurry b3) .: [:b2,{(succ b5)}:])) ) & ( for b5 being Ordinal of b1 holds
( b5 <> 0-element_of b1 & b5 is_limit_ordinal implies b4 . b5 = sup (b4 | b5) ) ) ) ) )
proof end;

theorem Th15: :: ZFREFLE1:15
for b1 being Ordinal
for b2 being Ordinal-Sequence holds
( b2 is increasing implies b1 +^ b2 is increasing )
proof end;

theorem Th16: :: ZFREFLE1:16
for b1, b2 being Ordinal
for b3 being Ordinal-Sequence holds (b1 +^ b3) | b2 = b1 +^ (b3 | b2)
proof end;

theorem Th17: :: ZFREFLE1:17
for b1 being Ordinal
for b2 being Ordinal-Sequence holds
( b2 is increasing & b2 is continuous implies b1 +^ b2 is continuous )
proof end;

definition
let c1, c2 be Ordinal;
pred c1 is_cofinal_with c2 means :: ZFREFLE1:def 5
ex b1 being Ordinal-Sequence st
( dom b1 = a2 & rng b1 c= a1 & b1 is increasing & a1 = sup b1 );
reflexivity
for b1 being Ordinal holds
ex b2 being Ordinal-Sequence st
( dom b2 = b1 & rng b2 c= b1 & b2 is increasing & b1 = sup b2 )
proof end;
end;

:: deftheorem Def5 defines is_cofinal_with ZFREFLE1:def 5 :
for b1, b2 being Ordinal holds
( b1 is_cofinal_with b2 iff ex b3 being Ordinal-Sequence st
( dom b3 = b2 & rng b3 c= b1 & b3 is increasing & b1 = sup b3 ) );

theorem Th18: :: ZFREFLE1:18
canceled;

theorem Th19: :: ZFREFLE1:19
for b1 being set
for b2 being Ordinal-Sequence holds
( b1 in rng b2 implies b1 is Ordinal )
proof end;

theorem Th20: :: ZFREFLE1:20
for b1 being Ordinal-Sequence holds rng b1 c= sup b1
proof end;

theorem Th21: :: ZFREFLE1:21
for b1, b2, b3 being Ordinal holds
( b1 is_cofinal_with b2 & b2 is_cofinal_with b3 implies b1 is_cofinal_with b3 )
proof end;

theorem Th22: :: ZFREFLE1:22
for b1, b2 being Ordinal holds
( b1 is_cofinal_with b2 implies b2 c= b1 )
proof end;

theorem Th23: :: ZFREFLE1:23
for b1, b2 being Ordinal holds
( b1 is_cofinal_with b2 & b2 is_cofinal_with b1 implies b1 = b2 )
proof end;

theorem Th24: :: ZFREFLE1:24
for b1 being Ordinal
for b2 being Ordinal-Sequence holds
( dom b2 <> {} & dom b2 is_limit_ordinal & b2 is increasing & b1 is_limes_of b2 implies b1 is_cofinal_with dom b2 )
proof end;

theorem Th25: :: ZFREFLE1:25
for b1 being Ordinal holds succ b1 is_cofinal_with 1
proof end;

theorem Th26: :: ZFREFLE1:26
for b1, b2 being Ordinal holds
not ( b1 is_cofinal_with succ b2 & ( for b3 being Ordinal holds
not b1 = succ b3 ) )
proof end;

theorem Th27: :: ZFREFLE1:27
for b1, b2 being Ordinal holds
( b1 is_cofinal_with b2 implies ( b1 is_limit_ordinal iff b2 is_limit_ordinal ) )
proof end;

theorem Th28: :: ZFREFLE1:28
for b1 being Ordinal holds
( b1 is_cofinal_with {} implies b1 = {} )
proof end;

theorem Th29: :: ZFREFLE1:29
for b1 being Universe
for b2 being Ordinal of b1 holds
not On b1 is_cofinal_with b2
proof end;

theorem Th30: :: ZFREFLE1:30
for b1 being Universe
for b2 being Ordinal of b1
for b3 being Ordinal-Sequence of b1 holds
not ( omega in b1 & b3 is increasing & b3 is continuous & ( for b4 being Ordinal of b1 holds
not ( b2 in b4 & b3 . b4 = b4 ) ) )
proof end;

theorem Th31: :: ZFREFLE1:31
for b1 being Universe
for b2 being Ordinal of b1
for b3 being Ordinal-Sequence of b1 holds
not ( omega in b1 & b3 is increasing & b3 is continuous & ( for b4 being Ordinal of b1 holds
not ( b2 in b4 & b3 . b4 = b4 & b4 is_cofinal_with omega ) ) )
proof end;

theorem Th32: :: ZFREFLE1:32
for b1 being Universe
for b2 being DOMAIN-Sequence of b1 holds
not ( omega in b1 & ( for b3, b4 being Ordinal of b1 holds
( b3 in b4 implies b2 . b3 c= b2 . b4 ) ) & ( for b3 being Ordinal of b1 holds
( b3 <> {} & b3 is_limit_ordinal implies b2 . b3 = Union (b2 | b3) ) ) & ( for b3 being Ordinal-Sequence of b1 holds
not ( b3 is increasing & b3 is continuous & ( for b4 being Ordinal of b1 holds
( b3 . b4 = b4 & {} <> b4 implies b2 . b4 is_elementary_subsystem_of Union b2 ) ) ) ) )
proof end;

theorem Th33: :: ZFREFLE1:33
for b1 being Universe
for b2 being Ordinal of b1 holds Rank b2 in b1
proof end;

theorem Th34: :: ZFREFLE1:34
for b1 being Universe
for b2 being Ordinal of b1 holds
( b2 <> {} implies Rank b2 is non empty Element of b1 )
proof end;

theorem Th35: :: ZFREFLE1:35
for b1 being Universe holds
not ( omega in b1 & ( for b2 being Ordinal-Sequence of b1 holds
not ( b2 is increasing & b2 is continuous & ( for b3 being Ordinal of b1
for b4 being non empty set holds
( b2 . b3 = b3 & {} <> b3 & b4 = Rank b3 implies b4 is_elementary_subsystem_of b1 ) ) ) ) )
proof end;

theorem Th36: :: ZFREFLE1:36
for b1 being Universe
for b2 being Ordinal of b1 holds
not ( omega in b1 & ( for b3 being Ordinal of b1
for b4 being non empty set holds
not ( b2 in b3 & b4 = Rank b3 & b4 is_elementary_subsystem_of b1 ) ) )
proof end;

theorem Th37: :: ZFREFLE1:37
for b1 being Universe holds
not ( omega in b1 & ( for b2 being Ordinal of b1
for b3 being non empty set holds
not ( b2 is_cofinal_with omega & b3 = Rank b2 & b3 is_elementary_subsystem_of b1 ) ) )
proof end;

theorem Th38: :: ZFREFLE1:38
for b1 being Universe
for b2 being DOMAIN-Sequence of b1 holds
not ( omega in b1 & ( for b3, b4 being Ordinal of b1 holds
( b3 in b4 implies b2 . b3 c= b2 . b4 ) ) & ( for b3 being Ordinal of b1 holds
( b3 <> {} & b3 is_limit_ordinal implies b2 . b3 = Union (b2 | b3) ) ) & ( for b3 being Ordinal-Sequence of b1 holds
not ( b3 is increasing & b3 is continuous & ( for b4 being Ordinal of b1 holds
( b3 . b4 = b4 & {} <> b4 implies b2 . b4 <==> Union b2 ) ) ) ) )
proof end;

theorem Th39: :: ZFREFLE1:39
for b1 being Universe holds
not ( omega in b1 & ( for b2 being Ordinal-Sequence of b1 holds
not ( b2 is increasing & b2 is continuous & ( for b3 being Ordinal of b1
for b4 being non empty set holds
( b2 . b3 = b3 & {} <> b3 & b4 = Rank b3 implies b4 <==> b1 ) ) ) ) )
proof end;

theorem Th40: :: ZFREFLE1:40
for b1 being Universe
for b2 being Ordinal of b1 holds
not ( omega in b1 & ( for b3 being Ordinal of b1
for b4 being non empty set holds
not ( b2 in b3 & b4 = Rank b3 & b4 <==> b1 ) ) )
proof end;

theorem Th41: :: ZFREFLE1:41
for b1 being Universe holds
not ( omega in b1 & ( for b2 being Ordinal of b1
for b3 being non empty set holds
not ( b2 is_cofinal_with omega & b3 = Rank b2 & b3 <==> b1 ) ) )
proof end;

theorem Th42: :: ZFREFLE1:42
for b1 being Universe holds
not ( omega in b1 & ( for b2 being Ordinal of b1
for b3 being non empty set holds
not ( b2 is_cofinal_with omega & b3 = Rank b2 & b3 is_a_model_of_ZF ) ) )
proof end;

theorem Th43: :: ZFREFLE1:43
for b1 being set
for b2 being Universe holds
not ( omega in b2 & b1 in b2 & ( for b3 being non empty set holds
not ( b1 in b3 & b3 in b2 & b3 is_a_model_of_ZF ) ) )
proof end;