Step |
Hyp |
Ref |
Expression |
1 |
|
ordtoplem.1 |
|- ( U. A e. On -> suc U. A e. S ) |
2 |
|
df-ne |
|- ( A =/= U. A <-> -. A = U. A ) |
3 |
|
ordeleqon |
|- ( Ord A <-> ( A e. On \/ A = On ) ) |
4 |
|
unon |
|- U. On = On |
5 |
4
|
eqcomi |
|- On = U. On |
6 |
|
id |
|- ( A = On -> A = On ) |
7 |
|
unieq |
|- ( A = On -> U. A = U. On ) |
8 |
5 6 7
|
3eqtr4a |
|- ( A = On -> A = U. A ) |
9 |
8
|
orim2i |
|- ( ( A e. On \/ A = On ) -> ( A e. On \/ A = U. A ) ) |
10 |
3 9
|
sylbi |
|- ( Ord A -> ( A e. On \/ A = U. A ) ) |
11 |
10
|
orcomd |
|- ( Ord A -> ( A = U. A \/ A e. On ) ) |
12 |
11
|
ord |
|- ( Ord A -> ( -. A = U. A -> A e. On ) ) |
13 |
|
orduniorsuc |
|- ( Ord A -> ( A = U. A \/ A = suc U. A ) ) |
14 |
13
|
ord |
|- ( Ord A -> ( -. A = U. A -> A = suc U. A ) ) |
15 |
|
onuni |
|- ( A e. On -> U. A e. On ) |
16 |
|
eleq1a |
|- ( suc U. A e. S -> ( A = suc U. A -> A e. S ) ) |
17 |
15 1 16
|
3syl |
|- ( A e. On -> ( A = suc U. A -> A e. S ) ) |
18 |
12 14 17
|
syl6c |
|- ( Ord A -> ( -. A = U. A -> A e. S ) ) |
19 |
2 18
|
syl5bi |
|- ( Ord A -> ( A =/= U. A -> A e. S ) ) |