Step |
Hyp |
Ref |
Expression |
1 |
|
cardon |
|- ( card ` A ) e. On |
2 |
|
cardon |
|- ( card ` B ) e. On |
3 |
|
oacl |
|- ( ( ( card ` A ) e. On /\ ( card ` B ) e. On ) -> ( ( card ` A ) +o ( card ` B ) ) e. On ) |
4 |
1 2 3
|
mp2an |
|- ( ( card ` A ) +o ( card ` B ) ) e. On |
5 |
|
cardadju |
|- ( ( A e. dom card /\ B e. dom card ) -> ( A |_| B ) ~~ ( ( card ` A ) +o ( card ` B ) ) ) |
6 |
5
|
ensymd |
|- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) +o ( card ` B ) ) ~~ ( A |_| B ) ) |
7 |
|
isnumi |
|- ( ( ( ( card ` A ) +o ( card ` B ) ) e. On /\ ( ( card ` A ) +o ( card ` B ) ) ~~ ( A |_| B ) ) -> ( A |_| B ) e. dom card ) |
8 |
4 6 7
|
sylancr |
|- ( ( A e. dom card /\ B e. dom card ) -> ( A |_| B ) e. dom card ) |