Step |
Hyp |
Ref |
Expression |
1 |
|
df-1o |
|- 1o = suc (/) |
2 |
1
|
fveq2i |
|- ( aleph ` 1o ) = ( aleph ` suc (/) ) |
3 |
|
alephsucpw |
|- ( aleph ` suc (/) ) ~<_ ~P ( aleph ` (/) ) |
4 |
|
fvex |
|- ( aleph ` (/) ) e. _V |
5 |
4
|
pw2en |
|- ~P ( aleph ` (/) ) ~~ ( 2o ^m ( aleph ` (/) ) ) |
6 |
|
domen2 |
|- ( ~P ( aleph ` (/) ) ~~ ( 2o ^m ( aleph ` (/) ) ) -> ( ( aleph ` suc (/) ) ~<_ ~P ( aleph ` (/) ) <-> ( aleph ` suc (/) ) ~<_ ( 2o ^m ( aleph ` (/) ) ) ) ) |
7 |
5 6
|
ax-mp |
|- ( ( aleph ` suc (/) ) ~<_ ~P ( aleph ` (/) ) <-> ( aleph ` suc (/) ) ~<_ ( 2o ^m ( aleph ` (/) ) ) ) |
8 |
3 7
|
mpbi |
|- ( aleph ` suc (/) ) ~<_ ( 2o ^m ( aleph ` (/) ) ) |
9 |
2 8
|
eqbrtri |
|- ( aleph ` 1o ) ~<_ ( 2o ^m ( aleph ` (/) ) ) |