Step |
Hyp |
Ref |
Expression |
1 |
|
alephgeom |
|- ( A e. On <-> _om C_ ( aleph ` A ) ) |
2 |
|
cardlim |
|- ( _om C_ ( card ` ( aleph ` A ) ) <-> Lim ( card ` ( aleph ` A ) ) ) |
3 |
|
alephcard |
|- ( card ` ( aleph ` A ) ) = ( aleph ` A ) |
4 |
3
|
sseq2i |
|- ( _om C_ ( card ` ( aleph ` A ) ) <-> _om C_ ( aleph ` A ) ) |
5 |
|
limeq |
|- ( ( card ` ( aleph ` A ) ) = ( aleph ` A ) -> ( Lim ( card ` ( aleph ` A ) ) <-> Lim ( aleph ` A ) ) ) |
6 |
3 5
|
ax-mp |
|- ( Lim ( card ` ( aleph ` A ) ) <-> Lim ( aleph ` A ) ) |
7 |
2 4 6
|
3bitr3i |
|- ( _om C_ ( aleph ` A ) <-> Lim ( aleph ` A ) ) |
8 |
1 7
|
bitri |
|- ( A e. On <-> Lim ( aleph ` A ) ) |