Step |
Hyp |
Ref |
Expression |
1 |
|
inf3lem.1 |
|- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) |
2 |
|
inf3lem.2 |
|- F = ( rec ( G , (/) ) |` _om ) |
3 |
|
inf3lem.3 |
|- A e. _V |
4 |
|
inf3lem.4 |
|- B e. _V |
5 |
1 2 3 4
|
inf3lemd |
|- ( A e. _om -> ( F ` A ) C_ x ) |
6 |
1 2 3 4
|
inf3lem2 |
|- ( ( x =/= (/) /\ x C_ U. x ) -> ( A e. _om -> ( F ` A ) =/= x ) ) |
7 |
6
|
com12 |
|- ( A e. _om -> ( ( x =/= (/) /\ x C_ U. x ) -> ( F ` A ) =/= x ) ) |
8 |
|
pssdifn0 |
|- ( ( ( F ` A ) C_ x /\ ( F ` A ) =/= x ) -> ( x \ ( F ` A ) ) =/= (/) ) |
9 |
5 7 8
|
syl6an |
|- ( A e. _om -> ( ( x =/= (/) /\ x C_ U. x ) -> ( x \ ( F ` A ) ) =/= (/) ) ) |
10 |
|
vex |
|- x e. _V |
11 |
10
|
difexi |
|- ( x \ ( F ` A ) ) e. _V |
12 |
|
zfreg |
|- ( ( ( x \ ( F ` A ) ) e. _V /\ ( x \ ( F ` A ) ) =/= (/) ) -> E. v e. ( x \ ( F ` A ) ) ( v i^i ( x \ ( F ` A ) ) ) = (/) ) |
13 |
11 12
|
mpan |
|- ( ( x \ ( F ` A ) ) =/= (/) -> E. v e. ( x \ ( F ` A ) ) ( v i^i ( x \ ( F ` A ) ) ) = (/) ) |
14 |
|
eldifi |
|- ( v e. ( x \ ( F ` A ) ) -> v e. x ) |
15 |
|
inssdif0 |
|- ( ( v i^i x ) C_ ( F ` A ) <-> ( v i^i ( x \ ( F ` A ) ) ) = (/) ) |
16 |
15
|
biimpri |
|- ( ( v i^i ( x \ ( F ` A ) ) ) = (/) -> ( v i^i x ) C_ ( F ` A ) ) |
17 |
14 16
|
anim12i |
|- ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> ( v e. x /\ ( v i^i x ) C_ ( F ` A ) ) ) |
18 |
|
vex |
|- v e. _V |
19 |
|
fvex |
|- ( F ` A ) e. _V |
20 |
1 2 18 19
|
inf3lema |
|- ( v e. ( G ` ( F ` A ) ) <-> ( v e. x /\ ( v i^i x ) C_ ( F ` A ) ) ) |
21 |
17 20
|
sylibr |
|- ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> v e. ( G ` ( F ` A ) ) ) |
22 |
1 2 3 4
|
inf3lemc |
|- ( A e. _om -> ( F ` suc A ) = ( G ` ( F ` A ) ) ) |
23 |
22
|
eleq2d |
|- ( A e. _om -> ( v e. ( F ` suc A ) <-> v e. ( G ` ( F ` A ) ) ) ) |
24 |
21 23
|
syl5ibr |
|- ( A e. _om -> ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> v e. ( F ` suc A ) ) ) |
25 |
|
eldifn |
|- ( v e. ( x \ ( F ` A ) ) -> -. v e. ( F ` A ) ) |
26 |
25
|
adantr |
|- ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> -. v e. ( F ` A ) ) |
27 |
24 26
|
jca2 |
|- ( A e. _om -> ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> ( v e. ( F ` suc A ) /\ -. v e. ( F ` A ) ) ) ) |
28 |
|
eleq2 |
|- ( ( F ` A ) = ( F ` suc A ) -> ( v e. ( F ` A ) <-> v e. ( F ` suc A ) ) ) |
29 |
28
|
biimprd |
|- ( ( F ` A ) = ( F ` suc A ) -> ( v e. ( F ` suc A ) -> v e. ( F ` A ) ) ) |
30 |
|
iman |
|- ( ( v e. ( F ` suc A ) -> v e. ( F ` A ) ) <-> -. ( v e. ( F ` suc A ) /\ -. v e. ( F ` A ) ) ) |
31 |
29 30
|
sylib |
|- ( ( F ` A ) = ( F ` suc A ) -> -. ( v e. ( F ` suc A ) /\ -. v e. ( F ` A ) ) ) |
32 |
31
|
necon2ai |
|- ( ( v e. ( F ` suc A ) /\ -. v e. ( F ` A ) ) -> ( F ` A ) =/= ( F ` suc A ) ) |
33 |
27 32
|
syl6 |
|- ( A e. _om -> ( ( v e. ( x \ ( F ` A ) ) /\ ( v i^i ( x \ ( F ` A ) ) ) = (/) ) -> ( F ` A ) =/= ( F ` suc A ) ) ) |
34 |
33
|
expd |
|- ( A e. _om -> ( v e. ( x \ ( F ` A ) ) -> ( ( v i^i ( x \ ( F ` A ) ) ) = (/) -> ( F ` A ) =/= ( F ` suc A ) ) ) ) |
35 |
34
|
rexlimdv |
|- ( A e. _om -> ( E. v e. ( x \ ( F ` A ) ) ( v i^i ( x \ ( F ` A ) ) ) = (/) -> ( F ` A ) =/= ( F ` suc A ) ) ) |
36 |
13 35
|
syl5 |
|- ( A e. _om -> ( ( x \ ( F ` A ) ) =/= (/) -> ( F ` A ) =/= ( F ` suc A ) ) ) |
37 |
9 36
|
syldc |
|- ( ( x =/= (/) /\ x C_ U. x ) -> ( A e. _om -> ( F ` A ) =/= ( F ` suc A ) ) ) |