Step |
Hyp |
Ref |
Expression |
1 |
|
ordom |
|- Ord _om |
2 |
|
ordelss |
|- ( ( Ord _om /\ A e. _om ) -> A C_ _om ) |
3 |
1 2
|
mpan |
|- ( A e. _om -> A C_ _om ) |
4 |
3
|
sspwd |
|- ( A e. _om -> ~P A C_ ~P _om ) |
5 |
4
|
sselda |
|- ( ( A e. _om /\ a e. ~P A ) -> a e. ~P _om ) |
6 |
|
nnfi |
|- ( A e. _om -> A e. Fin ) |
7 |
|
elpwi |
|- ( a e. ~P A -> a C_ A ) |
8 |
|
ssfi |
|- ( ( A e. Fin /\ a C_ A ) -> a e. Fin ) |
9 |
6 7 8
|
syl2an |
|- ( ( A e. _om /\ a e. ~P A ) -> a e. Fin ) |
10 |
5 9
|
elind |
|- ( ( A e. _om /\ a e. ~P A ) -> a e. ( ~P _om i^i Fin ) ) |
11 |
10
|
ex |
|- ( A e. _om -> ( a e. ~P A -> a e. ( ~P _om i^i Fin ) ) ) |
12 |
11
|
ssrdv |
|- ( A e. _om -> ~P A C_ ( ~P _om i^i Fin ) ) |