Step |
Hyp |
Ref |
Expression |
1 |
|
alephgeom |
|- ( A e. On <-> _om C_ ( aleph ` A ) ) |
2 |
|
fvex |
|- ( aleph ` A ) e. _V |
3 |
|
ssdomg |
|- ( ( aleph ` A ) e. _V -> ( _om C_ ( aleph ` A ) -> _om ~<_ ( aleph ` A ) ) ) |
4 |
2 3
|
ax-mp |
|- ( _om C_ ( aleph ` A ) -> _om ~<_ ( aleph ` A ) ) |
5 |
1 4
|
sylbi |
|- ( A e. On -> _om ~<_ ( aleph ` A ) ) |
6 |
|
alephon |
|- ( aleph ` A ) e. On |
7 |
|
onenon |
|- ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card ) |
8 |
6 7
|
ax-mp |
|- ( aleph ` A ) e. dom card |
9 |
5 8
|
jctil |
|- ( A e. On -> ( ( aleph ` A ) e. dom card /\ _om ~<_ ( aleph ` A ) ) ) |
10 |
|
alephgeom |
|- ( B e. On <-> _om C_ ( aleph ` B ) ) |
11 |
|
fvex |
|- ( aleph ` B ) e. _V |
12 |
|
ssdomg |
|- ( ( aleph ` B ) e. _V -> ( _om C_ ( aleph ` B ) -> _om ~<_ ( aleph ` B ) ) ) |
13 |
11 12
|
ax-mp |
|- ( _om C_ ( aleph ` B ) -> _om ~<_ ( aleph ` B ) ) |
14 |
|
infn0 |
|- ( _om ~<_ ( aleph ` B ) -> ( aleph ` B ) =/= (/) ) |
15 |
13 14
|
syl |
|- ( _om C_ ( aleph ` B ) -> ( aleph ` B ) =/= (/) ) |
16 |
10 15
|
sylbi |
|- ( B e. On -> ( aleph ` B ) =/= (/) ) |
17 |
|
alephon |
|- ( aleph ` B ) e. On |
18 |
|
onenon |
|- ( ( aleph ` B ) e. On -> ( aleph ` B ) e. dom card ) |
19 |
17 18
|
ax-mp |
|- ( aleph ` B ) e. dom card |
20 |
16 19
|
jctil |
|- ( B e. On -> ( ( aleph ` B ) e. dom card /\ ( aleph ` B ) =/= (/) ) ) |
21 |
|
infxp |
|- ( ( ( ( aleph ` A ) e. dom card /\ _om ~<_ ( aleph ` A ) ) /\ ( ( aleph ` B ) e. dom card /\ ( aleph ` B ) =/= (/) ) ) -> ( ( aleph ` A ) X. ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) ) |
22 |
9 20 21
|
syl2an |
|- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) X. ( aleph ` B ) ) ~~ ( ( aleph ` A ) u. ( aleph ` B ) ) ) |