Step |
Hyp |
Ref |
Expression |
1 |
|
nnon |
|- ( A e. _om -> A e. On ) |
2 |
|
nnon |
|- ( B e. _om -> B e. On ) |
3 |
|
onadju |
|- ( ( A e. On /\ B e. On ) -> ( A +o B ) ~~ ( A |_| B ) ) |
4 |
1 2 3
|
syl2an |
|- ( ( A e. _om /\ B e. _om ) -> ( A +o B ) ~~ ( A |_| B ) ) |
5 |
|
carden2b |
|- ( ( A +o B ) ~~ ( A |_| B ) -> ( card ` ( A +o B ) ) = ( card ` ( A |_| B ) ) ) |
6 |
4 5
|
syl |
|- ( ( A e. _om /\ B e. _om ) -> ( card ` ( A +o B ) ) = ( card ` ( A |_| B ) ) ) |
7 |
|
nnacl |
|- ( ( A e. _om /\ B e. _om ) -> ( A +o B ) e. _om ) |
8 |
|
cardnn |
|- ( ( A +o B ) e. _om -> ( card ` ( A +o B ) ) = ( A +o B ) ) |
9 |
7 8
|
syl |
|- ( ( A e. _om /\ B e. _om ) -> ( card ` ( A +o B ) ) = ( A +o B ) ) |
10 |
6 9
|
eqtr3d |
|- ( ( A e. _om /\ B e. _om ) -> ( card ` ( A |_| B ) ) = ( A +o B ) ) |