| Step |
Hyp |
Ref |
Expression |
| 1 |
|
2fveq3 |
|- ( x = (/) -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` (/) ) ) ) |
| 2 |
|
fveq2 |
|- ( x = (/) -> ( aleph ` x ) = ( aleph ` (/) ) ) |
| 3 |
1 2
|
eqeq12d |
|- ( x = (/) -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` (/) ) ) = ( aleph ` (/) ) ) ) |
| 4 |
|
2fveq3 |
|- ( x = y -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` y ) ) ) |
| 5 |
|
fveq2 |
|- ( x = y -> ( aleph ` x ) = ( aleph ` y ) ) |
| 6 |
4 5
|
eqeq12d |
|- ( x = y -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) ) |
| 7 |
|
2fveq3 |
|- ( x = suc y -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` suc y ) ) ) |
| 8 |
|
fveq2 |
|- ( x = suc y -> ( aleph ` x ) = ( aleph ` suc y ) ) |
| 9 |
7 8
|
eqeq12d |
|- ( x = suc y -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` suc y ) ) = ( aleph ` suc y ) ) ) |
| 10 |
|
2fveq3 |
|- ( x = A -> ( card ` ( aleph ` x ) ) = ( card ` ( aleph ` A ) ) ) |
| 11 |
|
fveq2 |
|- ( x = A -> ( aleph ` x ) = ( aleph ` A ) ) |
| 12 |
10 11
|
eqeq12d |
|- ( x = A -> ( ( card ` ( aleph ` x ) ) = ( aleph ` x ) <-> ( card ` ( aleph ` A ) ) = ( aleph ` A ) ) ) |
| 13 |
|
cardom |
|- ( card ` _om ) = _om |
| 14 |
|
aleph0 |
|- ( aleph ` (/) ) = _om |
| 15 |
14
|
fveq2i |
|- ( card ` ( aleph ` (/) ) ) = ( card ` _om ) |
| 16 |
13 15 14
|
3eqtr4i |
|- ( card ` ( aleph ` (/) ) ) = ( aleph ` (/) ) |
| 17 |
|
harcard |
|- ( card ` ( har ` ( aleph ` y ) ) ) = ( har ` ( aleph ` y ) ) |
| 18 |
|
alephsuc |
|- ( y e. On -> ( aleph ` suc y ) = ( har ` ( aleph ` y ) ) ) |
| 19 |
18
|
fveq2d |
|- ( y e. On -> ( card ` ( aleph ` suc y ) ) = ( card ` ( har ` ( aleph ` y ) ) ) ) |
| 20 |
17 19 18
|
3eqtr4a |
|- ( y e. On -> ( card ` ( aleph ` suc y ) ) = ( aleph ` suc y ) ) |
| 21 |
20
|
a1d |
|- ( y e. On -> ( ( card ` ( aleph ` y ) ) = ( aleph ` y ) -> ( card ` ( aleph ` suc y ) ) = ( aleph ` suc y ) ) ) |
| 22 |
|
cardiun |
|- ( x e. _V -> ( A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) -> ( card ` U_ y e. x ( aleph ` y ) ) = U_ y e. x ( aleph ` y ) ) ) |
| 23 |
22
|
elv |
|- ( A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) -> ( card ` U_ y e. x ( aleph ` y ) ) = U_ y e. x ( aleph ` y ) ) |
| 24 |
23
|
adantl |
|- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( card ` U_ y e. x ( aleph ` y ) ) = U_ y e. x ( aleph ` y ) ) |
| 25 |
|
vex |
|- x e. _V |
| 26 |
|
alephlim |
|- ( ( x e. _V /\ Lim x ) -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) ) |
| 27 |
25 26
|
mpan |
|- ( Lim x -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) ) |
| 28 |
27
|
adantr |
|- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( aleph ` x ) = U_ y e. x ( aleph ` y ) ) |
| 29 |
28
|
fveq2d |
|- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( card ` ( aleph ` x ) ) = ( card ` U_ y e. x ( aleph ` y ) ) ) |
| 30 |
24 29 28
|
3eqtr4d |
|- ( ( Lim x /\ A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) ) -> ( card ` ( aleph ` x ) ) = ( aleph ` x ) ) |
| 31 |
30
|
ex |
|- ( Lim x -> ( A. y e. x ( card ` ( aleph ` y ) ) = ( aleph ` y ) -> ( card ` ( aleph ` x ) ) = ( aleph ` x ) ) ) |
| 32 |
3 6 9 12 16 21 31
|
tfinds |
|- ( A e. On -> ( card ` ( aleph ` A ) ) = ( aleph ` A ) ) |
| 33 |
|
card0 |
|- ( card ` (/) ) = (/) |
| 34 |
|
alephfnon |
|- aleph Fn On |
| 35 |
34
|
fndmi |
|- dom aleph = On |
| 36 |
35
|
eleq2i |
|- ( A e. dom aleph <-> A e. On ) |
| 37 |
|
ndmfv |
|- ( -. A e. dom aleph -> ( aleph ` A ) = (/) ) |
| 38 |
36 37
|
sylnbir |
|- ( -. A e. On -> ( aleph ` A ) = (/) ) |
| 39 |
38
|
fveq2d |
|- ( -. A e. On -> ( card ` ( aleph ` A ) ) = ( card ` (/) ) ) |
| 40 |
33 39 38
|
3eqtr4a |
|- ( -. A e. On -> ( card ` ( aleph ` A ) ) = ( aleph ` A ) ) |
| 41 |
32 40
|
pm2.61i |
|- ( card ` ( aleph ` A ) ) = ( aleph ` A ) |