Step |
Hyp |
Ref |
Expression |
1 |
|
df-ne |
|- ( ( A ` x ) =/= ( B ` x ) <-> -. ( A ` x ) = ( B ` x ) ) |
2 |
1
|
rexbii |
|- ( E. x e. On ( A ` x ) =/= ( B ` x ) <-> E. x e. On -. ( A ` x ) = ( B ` x ) ) |
3 |
2
|
notbii |
|- ( -. E. x e. On ( A ` x ) =/= ( B ` x ) <-> -. E. x e. On -. ( A ` x ) = ( B ` x ) ) |
4 |
|
dfral2 |
|- ( A. x e. On ( A ` x ) = ( B ` x ) <-> -. E. x e. On -. ( A ` x ) = ( B ` x ) ) |
5 |
3 4
|
bitr4i |
|- ( -. E. x e. On ( A ` x ) =/= ( B ` x ) <-> A. x e. On ( A ` x ) = ( B ` x ) ) |
6 |
|
nodmord |
|- ( A e. No -> Ord dom A ) |
7 |
|
nodmord |
|- ( B e. No -> Ord dom B ) |
8 |
|
ordtri3or |
|- ( ( Ord dom A /\ Ord dom B ) -> ( dom A e. dom B \/ dom A = dom B \/ dom B e. dom A ) ) |
9 |
6 7 8
|
syl2an |
|- ( ( A e. No /\ B e. No ) -> ( dom A e. dom B \/ dom A = dom B \/ dom B e. dom A ) ) |
10 |
|
3orass |
|- ( ( dom A e. dom B \/ dom A = dom B \/ dom B e. dom A ) <-> ( dom A e. dom B \/ ( dom A = dom B \/ dom B e. dom A ) ) ) |
11 |
|
or12 |
|- ( ( dom A e. dom B \/ ( dom A = dom B \/ dom B e. dom A ) ) <-> ( dom A = dom B \/ ( dom A e. dom B \/ dom B e. dom A ) ) ) |
12 |
10 11
|
bitri |
|- ( ( dom A e. dom B \/ dom A = dom B \/ dom B e. dom A ) <-> ( dom A = dom B \/ ( dom A e. dom B \/ dom B e. dom A ) ) ) |
13 |
9 12
|
sylib |
|- ( ( A e. No /\ B e. No ) -> ( dom A = dom B \/ ( dom A e. dom B \/ dom B e. dom A ) ) ) |
14 |
13
|
ord |
|- ( ( A e. No /\ B e. No ) -> ( -. dom A = dom B -> ( dom A e. dom B \/ dom B e. dom A ) ) ) |
15 |
|
noseponlem |
|- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> -. A. x e. On ( A ` x ) = ( B ` x ) ) |
16 |
15
|
3expia |
|- ( ( A e. No /\ B e. No ) -> ( dom A e. dom B -> -. A. x e. On ( A ` x ) = ( B ` x ) ) ) |
17 |
|
noseponlem |
|- ( ( B e. No /\ A e. No /\ dom B e. dom A ) -> -. A. x e. On ( B ` x ) = ( A ` x ) ) |
18 |
|
eqcom |
|- ( ( A ` x ) = ( B ` x ) <-> ( B ` x ) = ( A ` x ) ) |
19 |
18
|
ralbii |
|- ( A. x e. On ( A ` x ) = ( B ` x ) <-> A. x e. On ( B ` x ) = ( A ` x ) ) |
20 |
17 19
|
sylnibr |
|- ( ( B e. No /\ A e. No /\ dom B e. dom A ) -> -. A. x e. On ( A ` x ) = ( B ` x ) ) |
21 |
20
|
3expia |
|- ( ( B e. No /\ A e. No ) -> ( dom B e. dom A -> -. A. x e. On ( A ` x ) = ( B ` x ) ) ) |
22 |
21
|
ancoms |
|- ( ( A e. No /\ B e. No ) -> ( dom B e. dom A -> -. A. x e. On ( A ` x ) = ( B ` x ) ) ) |
23 |
16 22
|
jaod |
|- ( ( A e. No /\ B e. No ) -> ( ( dom A e. dom B \/ dom B e. dom A ) -> -. A. x e. On ( A ` x ) = ( B ` x ) ) ) |
24 |
14 23
|
syld |
|- ( ( A e. No /\ B e. No ) -> ( -. dom A = dom B -> -. A. x e. On ( A ` x ) = ( B ` x ) ) ) |
25 |
24
|
con4d |
|- ( ( A e. No /\ B e. No ) -> ( A. x e. On ( A ` x ) = ( B ` x ) -> dom A = dom B ) ) |
26 |
25
|
3impia |
|- ( ( A e. No /\ B e. No /\ A. x e. On ( A ` x ) = ( B ` x ) ) -> dom A = dom B ) |
27 |
|
ordsson |
|- ( Ord dom A -> dom A C_ On ) |
28 |
|
ssralv |
|- ( dom A C_ On -> ( A. x e. On ( A ` x ) = ( B ` x ) -> A. x e. dom A ( A ` x ) = ( B ` x ) ) ) |
29 |
6 27 28
|
3syl |
|- ( A e. No -> ( A. x e. On ( A ` x ) = ( B ` x ) -> A. x e. dom A ( A ` x ) = ( B ` x ) ) ) |
30 |
29
|
adantr |
|- ( ( A e. No /\ B e. No ) -> ( A. x e. On ( A ` x ) = ( B ` x ) -> A. x e. dom A ( A ` x ) = ( B ` x ) ) ) |
31 |
30
|
3impia |
|- ( ( A e. No /\ B e. No /\ A. x e. On ( A ` x ) = ( B ` x ) ) -> A. x e. dom A ( A ` x ) = ( B ` x ) ) |
32 |
|
nofun |
|- ( A e. No -> Fun A ) |
33 |
32
|
3ad2ant1 |
|- ( ( A e. No /\ B e. No /\ A. x e. On ( A ` x ) = ( B ` x ) ) -> Fun A ) |
34 |
|
nofun |
|- ( B e. No -> Fun B ) |
35 |
34
|
3ad2ant2 |
|- ( ( A e. No /\ B e. No /\ A. x e. On ( A ` x ) = ( B ` x ) ) -> Fun B ) |
36 |
|
eqfunfv |
|- ( ( Fun A /\ Fun B ) -> ( A = B <-> ( dom A = dom B /\ A. x e. dom A ( A ` x ) = ( B ` x ) ) ) ) |
37 |
33 35 36
|
syl2anc |
|- ( ( A e. No /\ B e. No /\ A. x e. On ( A ` x ) = ( B ` x ) ) -> ( A = B <-> ( dom A = dom B /\ A. x e. dom A ( A ` x ) = ( B ` x ) ) ) ) |
38 |
26 31 37
|
mpbir2and |
|- ( ( A e. No /\ B e. No /\ A. x e. On ( A ` x ) = ( B ` x ) ) -> A = B ) |
39 |
38
|
3expia |
|- ( ( A e. No /\ B e. No ) -> ( A. x e. On ( A ` x ) = ( B ` x ) -> A = B ) ) |
40 |
5 39
|
syl5bi |
|- ( ( A e. No /\ B e. No ) -> ( -. E. x e. On ( A ` x ) =/= ( B ` x ) -> A = B ) ) |
41 |
40
|
necon1ad |
|- ( ( A e. No /\ B e. No ) -> ( A =/= B -> E. x e. On ( A ` x ) =/= ( B ` x ) ) ) |
42 |
41
|
3impia |
|- ( ( A e. No /\ B e. No /\ A =/= B ) -> E. x e. On ( A ` x ) =/= ( B ` x ) ) |
43 |
|
onintrab2 |
|- ( E. x e. On ( A ` x ) =/= ( B ` x ) <-> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. On ) |
44 |
42 43
|
sylib |
|- ( ( A e. No /\ B e. No /\ A =/= B ) -> |^| { x e. On | ( A ` x ) =/= ( B ` x ) } e. On ) |