Step |
Hyp |
Ref |
Expression |
1 |
|
unieq |
|- ( S = (/) -> U. S = U. (/) ) |
2 |
|
uni0 |
|- U. (/) = (/) |
3 |
|
peano1 |
|- (/) e. _om |
4 |
2 3
|
eqeltri |
|- U. (/) e. _om |
5 |
1 4
|
eqeltrdi |
|- ( S = (/) -> U. S e. _om ) |
6 |
5
|
adantl |
|- ( ( ( S C_ _om /\ S e. Fin ) /\ S = (/) ) -> U. S e. _om ) |
7 |
|
simpll |
|- ( ( ( S C_ _om /\ S e. Fin ) /\ S =/= (/) ) -> S C_ _om ) |
8 |
|
omsson |
|- _om C_ On |
9 |
7 8
|
sstrdi |
|- ( ( ( S C_ _om /\ S e. Fin ) /\ S =/= (/) ) -> S C_ On ) |
10 |
|
simplr |
|- ( ( ( S C_ _om /\ S e. Fin ) /\ S =/= (/) ) -> S e. Fin ) |
11 |
|
simpr |
|- ( ( ( S C_ _om /\ S e. Fin ) /\ S =/= (/) ) -> S =/= (/) ) |
12 |
|
ordunifi |
|- ( ( S C_ On /\ S e. Fin /\ S =/= (/) ) -> U. S e. S ) |
13 |
9 10 11 12
|
syl3anc |
|- ( ( ( S C_ _om /\ S e. Fin ) /\ S =/= (/) ) -> U. S e. S ) |
14 |
7 13
|
sseldd |
|- ( ( ( S C_ _om /\ S e. Fin ) /\ S =/= (/) ) -> U. S e. _om ) |
15 |
6 14
|
pm2.61dane |
|- ( ( S C_ _om /\ S e. Fin ) -> U. S e. _om ) |