:: ZFMODEL1 semantic presentation
set c1 = x. 0;
set c2 = x. 1;
set c3 = x. 2;
set c4 = x. 3;
set c5 = x. 4;
theorem Th1: :: ZFMODEL1:1
theorem Th2: :: ZFMODEL1:2
theorem Th3: :: ZFMODEL1:3
theorem Th4: :: ZFMODEL1:4
theorem Th5: :: ZFMODEL1:5
theorem Th6: :: ZFMODEL1:6
theorem Th7: :: ZFMODEL1:7
theorem Th8: :: ZFMODEL1:8
theorem Th9: :: ZFMODEL1:9
defpred S1[ Nat] means for b1 being Variable
for b2 being non empty set
for b3 being ZF-formula
for b4 being Function of VAR ,b2 holds
( len b3 = a1 & not b1 in Free b3 & b2,b4 |= b3 implies b2,b4 |= All b1,b3 );
Lemma5:
for b1 being Nat holds
( ( for b2 being Nat holds
( b2 < b1 implies S1[b2] ) ) implies S1[b1] )
theorem Th10: :: ZFMODEL1:10
Lemma7:
for b1 being Variable
for b2 being ZF-formula holds
( the_scope_of (All b1,b2) = b2 & bound_in (All b1,b2) = b1 )
theorem Th11: :: ZFMODEL1:11
theorem Th12: :: ZFMODEL1:12
for b
1, b
2, b
3 being
Variablefor b
4 being
ZF-formulafor b
5 being non
empty set for b
6 being
Function of
VAR ,b
5 holds
(
{b1,b2,b3} misses Free b
4 & b
5,b
6 |= b
4 implies b
5,b
6 |= All b
1,b
2,b
3,b
4 )
definition
let c
6 be
ZF-formula;
let c
7 be non
empty set ;
let c
8 be
Function of
VAR ,c
7;
assume E9:
( not
x. 0
in Free c
6 & c
7,c
8 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(c6 <=> ((x. 4) '=' (x. 0))))) )
;
func def_func' c
1,c
3 -> Function of a
2,a
2 means :
Def1:
:: ZFMODEL1:def 1
for b
1 being
Function of
VAR ,a
2 holds
( ( for b
2 being
Variable holds
not ( b
1 . b
2 <> a
3 . b
2 & not
x. 0
= b
2 & not
x. 3
= b
2 & not
x. 4
= b
2 ) ) implies ( a
2,b
1 |= a
1 iff a
4 . (b1 . (x. 3)) = b
1 . (x. 4) ) );
existence
ex b1 being Function of c7,c7 st
for b2 being Function of VAR ,c7 holds
( ( for b3 being Variable holds
not ( b2 . b3 <> c8 . b3 & not x. 0 = b3 & not x. 3 = b3 & not x. 4 = b3 ) ) implies ( c7,b2 |= c6 iff b1 . (b2 . (x. 3)) = b2 . (x. 4) ) )
uniqueness
for b1, b2 being Function of c7,c7 holds
( ( for b3 being Function of VAR ,c7 holds
( ( for b4 being Variable holds
not ( b3 . b4 <> c8 . b4 & not x. 0 = b4 & not x. 3 = b4 & not x. 4 = b4 ) ) implies ( c7,b3 |= c6 iff b1 . (b3 . (x. 3)) = b3 . (x. 4) ) ) ) & ( for b3 being Function of VAR ,c7 holds
( ( for b4 being Variable holds
not ( b3 . b4 <> c8 . b4 & not x. 0 = b4 & not x. 3 = b4 & not x. 4 = b4 ) ) implies ( c7,b3 |= c6 iff b2 . (b3 . (x. 3)) = b3 . (x. 4) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines def_func' ZFMODEL1:def 1 :
theorem Th13: :: ZFMODEL1:13
canceled;
theorem Th14: :: ZFMODEL1:14
definition
let c
6 be
ZF-formula;
let c
7 be non
empty set ;
assume E11:
(
Free c
6 c= {(x. 3),(x. 4)} & c
7 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(c6 <=> ((x. 4) '=' (x. 0))))) )
;
func def_func c
1,c
2 -> Function of a
2,a
2 means :
Def2:
:: ZFMODEL1:def 2
for b
1 being
Function of
VAR ,a
2 holds
( a
2,b
1 |= a
1 iff a
3 . (b1 . (x. 3)) = b
1 . (x. 4) );
existence
ex b1 being Function of c7,c7 st
for b2 being Function of VAR ,c7 holds
( c7,b2 |= c6 iff b1 . (b2 . (x. 3)) = b2 . (x. 4) )
uniqueness
for b1, b2 being Function of c7,c7 holds
( ( for b3 being Function of VAR ,c7 holds
( c7,b3 |= c6 iff b1 . (b3 . (x. 3)) = b3 . (x. 4) ) ) & ( for b3 being Function of VAR ,c7 holds
( c7,b3 |= c6 iff b2 . (b3 . (x. 3)) = b3 . (x. 4) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines def_func ZFMODEL1:def 2 :
definition
let c
6 be
Function;
let c
7 be non
empty set ;
pred c
1 is_definable_in c
2 means :: ZFMODEL1:def 3
ex b
1 being
ZF-formula st
(
Free b
1 c= {(x. 3),(x. 4)} & a
2 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b1 <=> ((x. 4) '=' (x. 0))))) & a
1 = def_func b
1,a
2 );
pred c
1 is_parametrically_definable_in c
2 means :
Def4:
:: ZFMODEL1:def 4
ex b
1 being
ZF-formulaex b
2 being
Function of
VAR ,a
2 st
(
{(x. 0),(x. 1),(x. 2)} misses Free b
1 & a
2,b
2 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b1 <=> ((x. 4) '=' (x. 0))))) & a
1 = def_func' b
1,b
2 );
end;
:: deftheorem Def3 defines is_definable_in ZFMODEL1:def 3 :
:: deftheorem Def4 defines is_parametrically_definable_in ZFMODEL1:def 4 :
theorem Th15: :: ZFMODEL1:15
canceled;
theorem Th16: :: ZFMODEL1:16
canceled;
theorem Th17: :: ZFMODEL1:17
canceled;
theorem Th18: :: ZFMODEL1:18
theorem Th19: :: ZFMODEL1:19
for b
1 being non
empty set holds
( b
1 is
epsilon-transitive implies ( ( for b
2 being
ZF-formula holds
(
{(x. 0),(x. 1),(x. 2)} misses Free b
2 implies b
1 |= the_axiom_of_substitution_for b
2 ) ) iff for b
2 being
ZF-formulafor b
3 being
Function of
VAR ,b
1 holds
(
{(x. 0),(x. 1),(x. 2)} misses Free b
2 & b
1,b
3 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b2 <=> ((x. 4) '=' (x. 0))))) implies for b
4 being
Element of b
1 holds
(def_func' b2,b3) .: b
4 in b
1 ) ) )
theorem Th20: :: ZFMODEL1:20
Lemma14:
for b1 being non empty set holds
( b1 is epsilon-transitive implies for b2, b3 being Element of b1 holds
( ( for b4 being Element of b1 holds
( b4 in b2 iff b4 in b3 ) ) implies b2 = b3 ) )
theorem Th21: :: ZFMODEL1:21
for b
1 being non
empty set holds
( b
1 is_a_model_of_ZF implies ( b
1 is
epsilon-transitive & ( for b
2, b
3 being
Element of b
1 holds
( ( for b
4 being
Element of b
1 holds
( b
4 in b
2 iff b
4 in b
3 ) ) implies b
2 = b
3 ) ) & ( for b
2, b
3 being
Element of b
1 holds
{b2,b3} in b
1 ) & ( for b
2 being
Element of b
1 holds
union b
2 in b
1 ) & ex b
2 being
Element of b
1 st
( b
2 <> {} & ( for b
3 being
Element of b
1 holds
not ( b
3 in b
2 & ( for b
4 being
Element of b
1 holds
not ( b
3 c< b
4 & b
4 in b
2 ) ) ) ) ) & ( for b
2 being
Element of b
1 holds b
1 /\ (bool b2) in b
1 ) & ( for b
2 being
ZF-formulafor b
3 being
Function of
VAR ,b
1 holds
(
{(x. 0),(x. 1),(x. 2)} misses Free b
2 & b
1,b
3 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b2 <=> ((x. 4) '=' (x. 0))))) implies for b
4 being
Element of b
1 holds
(def_func' b2,b3) .: b
4 in b
1 ) ) ) )
theorem Th22: :: ZFMODEL1:22
for b
1 being non
empty set holds
( b
1 is
epsilon-transitive & ( for b
2, b
3 being
Element of b
1 holds
{b2,b3} in b
1 ) & ( for b
2 being
Element of b
1 holds
union b
2 in b
1 ) & ex b
2 being
Element of b
1 st
( b
2 <> {} & ( for b
3 being
Element of b
1 holds
not ( b
3 in b
2 & ( for b
4 being
Element of b
1 holds
not ( b
3 c< b
4 & b
4 in b
2 ) ) ) ) ) & ( for b
2 being
Element of b
1 holds b
1 /\ (bool b2) in b
1 ) & ( for b
2 being
ZF-formulafor b
3 being
Function of
VAR ,b
1 holds
(
{(x. 0),(x. 1),(x. 2)} misses Free b
2 & b
1,b
3 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b2 <=> ((x. 4) '=' (x. 0))))) implies for b
4 being
Element of b
1 holds
(def_func' b2,b3) .: b
4 in b
1 ) ) implies b
1 is_a_model_of_ZF )