:: ZFMODEL2 semantic presentation
theorem Th1: :: ZFMODEL2:1
theorem Th2: :: ZFMODEL2:2
theorem Th3: :: ZFMODEL2:3
theorem Th4: :: ZFMODEL2:4
theorem Th5: :: ZFMODEL2:5
theorem Th6: :: ZFMODEL2:6
theorem Th7: :: ZFMODEL2:7
for b
1, b
2, b
3 being
Variablefor b
4 being non
empty set for b
5, b
6, b
7 being
Element of b
4for b
8 being
Function of
VAR ,b
4 holds
( b
1 <> b
2 & b
2 <> b
3 & b
3 <> b
1 implies (
((b8 / b1,b5) / b2,b6) / b
3,b
7 = ((b8 / b3,b7) / b2,b6) / b
1,b
5 &
((b8 / b1,b5) / b2,b6) / b
3,b
7 = ((b8 / b2,b6) / b3,b7) / b
1,b
5 ) )
theorem Th8: :: ZFMODEL2:8
for b
1, b
2, b
3, b
4 being
Variablefor b
5 being non
empty set for b
6, b
7, b
8, b
9 being
Element of b
5for b
10 being
Function of
VAR ,b
5 holds
( b
1 <> b
2 & b
1 <> b
3 & b
1 <> b
4 & b
2 <> b
3 & b
2 <> b
4 & b
3 <> b
4 implies (
(((b10 / b1,b6) / b2,b7) / b3,b8) / b
4,b
9 = (((b10 / b2,b7) / b3,b8) / b4,b9) / b
1,b
6 &
(((b10 / b1,b6) / b2,b7) / b3,b8) / b
4,b
9 = (((b10 / b3,b8) / b4,b9) / b1,b6) / b
2,b
7 &
(((b10 / b1,b6) / b2,b7) / b3,b8) / b
4,b
9 = (((b10 / b4,b9) / b2,b7) / b3,b8) / b
1,b
6 ) )
theorem Th9: :: ZFMODEL2:9
for b
1, b
2, b
3, b
4 being
Variablefor b
5 being non
empty set for b
6, b
7, b
8, b
9, b
10 being
Element of b
5for b
11 being
Function of
VAR ,b
5 holds
(
((b11 / b1,b6) / b2,b7) / b
1,b
8 = (b11 / b2,b7) / b
1,b
8 &
(((b11 / b1,b6) / b2,b7) / b3,b9) / b
1,b
8 = ((b11 / b2,b7) / b3,b9) / b
1,b
8 &
((((b11 / b1,b6) / b2,b7) / b3,b9) / b4,b10) / b
1,b
8 = (((b11 / b2,b7) / b3,b9) / b4,b10) / b
1,b
8 )
theorem Th10: :: ZFMODEL2:10
theorem Th11: :: ZFMODEL2:11
for b
1 being non
empty set for b
2 being
ZF-formulafor b
3 being
Function of
VAR ,b
1 holds
( not
x. 0
in 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, b
5 being
Element of b
1 holds
(
(def_func' b2,b3) . b
4 = b
5 iff b
1,
(b3 / (x. 3),b4) / (x. 4),b
5 |= b
2 ) )
Lemma11:
( x. 0 <> x. 3 & x. 0 <> x. 4 & x. 3 <> x. 4 )
by ZF_LANG1:86;
theorem Th12: :: ZFMODEL2:12
theorem Th13: :: ZFMODEL2:13
theorem Th14: :: ZFMODEL2:14
theorem Th15: :: ZFMODEL2:15
for b
1, b
2 being
Variablefor b
3 being non
empty set for b
4 being
ZF-formulafor b
5 being
Function of
VAR ,b
3 holds
( not
x. 0
in Free b
4 & b
3,b
5 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b4 <=> ((x. 4) '=' (x. 0))))) & not b
1 in variables_in b
4 & b
2 <> x. 3 & b
2 <> x. 4 & not b
2 in Free b
4 & b
1 <> x. 0 & b
1 <> x. 3 & b
1 <> x. 4 implies ( not
x. 0
in Free (b4 / b2,b1) & b
3,b
5 / b
1,
(b5 . b2) |= All (x. 3),
(Ex (x. 0),(All (x. 4),((b4 / b2,b1) <=> ((x. 4) '=' (x. 0))))) &
def_func' b
4,b
5 = def_func' (b4 / b2,b1),
(b5 / b1,(b5 . b2)) ) )
theorem Th16: :: ZFMODEL2:16
theorem Th17: :: ZFMODEL2:17
for b
1 being non
empty set for b
2 being
Natfor b
3 being
ZF-formulafor b
4 being
Function of
VAR ,b
1 holds
not ( not
x. 0
in Free b
3 & b
1,b
4 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b3 <=> ((x. 4) '=' (x. 0))))) & ( for b
5 being
ZF-formulafor b
6 being
Function of
VAR ,b
1 holds
not ( ( for b
7 being
Nat holds
not ( b
7 < b
2 &
x. b
7 in variables_in b
5 & not b
7 = 3 & not b
7 = 4 ) ) & not
x. 0
in Free b
5 & b
1,b
6 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b5 <=> ((x. 4) '=' (x. 0))))) &
def_func' b
3,b
4 = def_func' b
5,b
6 ) ) )
theorem Th18: :: ZFMODEL2:18
for b
1 being non
empty set for b
2 being
ZF-formulafor b
3 being
Function of
VAR ,b
1 holds
not ( not
x. 0
in Free b
2 & b
1,b
3 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b2 <=> ((x. 4) '=' (x. 0))))) & ( for b
4 being
ZF-formulafor b
5 being
Function of
VAR ,b
1 holds
not (
(Free b2) /\ (Free b4) c= {(x. 3),(x. 4)} & not
x. 0
in Free b
4 & b
1,b
5 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b4 <=> ((x. 4) '=' (x. 0))))) &
def_func' b
2,b
3 = def_func' b
4,b
5 ) ) )
theorem Th19: :: ZFMODEL2:19
theorem Th20: :: ZFMODEL2:20
theorem Th21: :: ZFMODEL2:21
theorem Th22: :: ZFMODEL2:22
theorem Th23: :: ZFMODEL2:23
for b
1 being non
empty set for b
2, b
3, b
4 being
ZF-formulafor b
5 being
Function of
VAR ,b
1 holds
(
{(x. 0),(x. 1),(x. 2)} misses Free b
2 & b
1,b
5 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b2 <=> ((x. 4) '=' (x. 0))))) &
{(x. 0),(x. 1),(x. 2)} misses Free b
3 & b
1,b
5 |= All (x. 3),
(Ex (x. 0),(All (x. 4),(b3 <=> ((x. 4) '=' (x. 0))))) &
{(x. 0),(x. 1),(x. 2)} misses Free b
4 & not
x. 4
in Free b
4 implies for b
6 being
Function holds
(
dom b
6 = b
1 & ( for b
7 being
Element of b
1 holds
( ( b
1,b
5 / (x. 3),b
7 |= b
4 implies b
6 . b
7 = (def_func' b2,b5) . b
7 ) & ( b
1,b
5 / (x. 3),b
7 |= 'not' b
4 implies b
6 . b
7 = (def_func' b3,b5) . b
7 ) ) ) implies b
6 is_parametrically_definable_in b
1 ) )
theorem Th24: :: ZFMODEL2:24
theorem Th25: :: ZFMODEL2:25