Step |
Hyp |
Ref |
Expression |
1 |
|
inawina |
|- ( x e. Inacc -> x e. InaccW ) |
2 |
|
winaon |
|- ( x e. InaccW -> x e. On ) |
3 |
1 2
|
syl |
|- ( x e. Inacc -> x e. On ) |
4 |
3
|
ssriv |
|- Inacc C_ On |
5 |
|
onmindif |
|- ( ( Inacc C_ On /\ A e. On ) -> A e. |^| ( Inacc \ suc A ) ) |
6 |
4 5
|
mpan |
|- ( A e. On -> A e. |^| ( Inacc \ suc A ) ) |
7 |
6
|
adantr |
|- ( ( A e. On /\ x = |^| ( Inacc \ suc A ) ) -> A e. |^| ( Inacc \ suc A ) ) |
8 |
|
simpr |
|- ( ( A e. On /\ x = |^| ( Inacc \ suc A ) ) -> x = |^| ( Inacc \ suc A ) ) |
9 |
7 8
|
eleqtrrd |
|- ( ( A e. On /\ x = |^| ( Inacc \ suc A ) ) -> A e. x ) |
10 |
|
difss |
|- ( Inacc \ suc A ) C_ Inacc |
11 |
10 4
|
sstri |
|- ( Inacc \ suc A ) C_ On |
12 |
|
inaprc |
|- Inacc e/ _V |
13 |
12
|
neli |
|- -. Inacc e. _V |
14 |
|
ssdif0 |
|- ( Inacc C_ suc A <-> ( Inacc \ suc A ) = (/) ) |
15 |
|
sucexg |
|- ( A e. On -> suc A e. _V ) |
16 |
|
ssexg |
|- ( ( Inacc C_ suc A /\ suc A e. _V ) -> Inacc e. _V ) |
17 |
16
|
expcom |
|- ( suc A e. _V -> ( Inacc C_ suc A -> Inacc e. _V ) ) |
18 |
15 17
|
syl |
|- ( A e. On -> ( Inacc C_ suc A -> Inacc e. _V ) ) |
19 |
14 18
|
syl5bir |
|- ( A e. On -> ( ( Inacc \ suc A ) = (/) -> Inacc e. _V ) ) |
20 |
13 19
|
mtoi |
|- ( A e. On -> -. ( Inacc \ suc A ) = (/) ) |
21 |
20
|
neqned |
|- ( A e. On -> ( Inacc \ suc A ) =/= (/) ) |
22 |
|
onint |
|- ( ( ( Inacc \ suc A ) C_ On /\ ( Inacc \ suc A ) =/= (/) ) -> |^| ( Inacc \ suc A ) e. ( Inacc \ suc A ) ) |
23 |
11 21 22
|
sylancr |
|- ( A e. On -> |^| ( Inacc \ suc A ) e. ( Inacc \ suc A ) ) |
24 |
23
|
eldifad |
|- ( A e. On -> |^| ( Inacc \ suc A ) e. Inacc ) |
25 |
9 24
|
rspcime |
|- ( A e. On -> E. x e. Inacc A e. x ) |