Step |
Hyp |
Ref |
Expression |
1 |
|
lrrec.1 |
|- R = { <. x , y >. | x e. ( ( _L ` y ) u. ( _R ` y ) ) } |
2 |
1
|
lrrecval |
|- ( ( A e. No /\ B e. No ) -> ( A R B <-> A e. ( ( _L ` B ) u. ( _R ` B ) ) ) ) |
3 |
|
lrold |
|- ( ( _L ` B ) u. ( _R ` B ) ) = ( _Old ` ( bday ` B ) ) |
4 |
3
|
a1i |
|- ( ( A e. No /\ B e. No ) -> ( ( _L ` B ) u. ( _R ` B ) ) = ( _Old ` ( bday ` B ) ) ) |
5 |
4
|
eleq2d |
|- ( ( A e. No /\ B e. No ) -> ( A e. ( ( _L ` B ) u. ( _R ` B ) ) <-> A e. ( _Old ` ( bday ` B ) ) ) ) |
6 |
|
bdayelon |
|- ( bday ` B ) e. On |
7 |
|
oldbday |
|- ( ( ( bday ` B ) e. On /\ A e. No ) -> ( A e. ( _Old ` ( bday ` B ) ) <-> ( bday ` A ) e. ( bday ` B ) ) ) |
8 |
6 7
|
mpan |
|- ( A e. No -> ( A e. ( _Old ` ( bday ` B ) ) <-> ( bday ` A ) e. ( bday ` B ) ) ) |
9 |
8
|
adantr |
|- ( ( A e. No /\ B e. No ) -> ( A e. ( _Old ` ( bday ` B ) ) <-> ( bday ` A ) e. ( bday ` B ) ) ) |
10 |
2 5 9
|
3bitrd |
|- ( ( A e. No /\ B e. No ) -> ( A R B <-> ( bday ` A ) e. ( bday ` B ) ) ) |