Step |
Hyp |
Ref |
Expression |
1 |
|
df-nel |
|- ( A e/ _V <-> -. A e. _V ) |
2 |
|
ssorduni |
|- ( A C_ On -> Ord U. A ) |
3 |
|
ordeleqon |
|- ( Ord U. A <-> ( U. A e. On \/ U. A = On ) ) |
4 |
2 3
|
sylib |
|- ( A C_ On -> ( U. A e. On \/ U. A = On ) ) |
5 |
4
|
orcomd |
|- ( A C_ On -> ( U. A = On \/ U. A e. On ) ) |
6 |
5
|
ord |
|- ( A C_ On -> ( -. U. A = On -> U. A e. On ) ) |
7 |
|
uniexr |
|- ( U. A e. On -> A e. _V ) |
8 |
6 7
|
syl6 |
|- ( A C_ On -> ( -. U. A = On -> A e. _V ) ) |
9 |
8
|
con1d |
|- ( A C_ On -> ( -. A e. _V -> U. A = On ) ) |
10 |
|
onprc |
|- -. On e. _V |
11 |
|
uniexg |
|- ( A e. _V -> U. A e. _V ) |
12 |
|
eleq1 |
|- ( U. A = On -> ( U. A e. _V <-> On e. _V ) ) |
13 |
11 12
|
syl5ib |
|- ( U. A = On -> ( A e. _V -> On e. _V ) ) |
14 |
10 13
|
mtoi |
|- ( U. A = On -> -. A e. _V ) |
15 |
9 14
|
impbid1 |
|- ( A C_ On -> ( -. A e. _V <-> U. A = On ) ) |
16 |
1 15
|
syl5bb |
|- ( A C_ On -> ( A e/ _V <-> U. A = On ) ) |