Step |
Hyp |
Ref |
Expression |
1 |
|
simprr |
|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> B ~<_ ~P A ) |
2 |
|
brdom2 |
|- ( B ~<_ ~P A <-> ( B ~< ~P A \/ B ~~ ~P A ) ) |
3 |
1 2
|
sylib |
|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( B ~< ~P A \/ B ~~ ~P A ) ) |
4 |
|
gchen1 |
|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> A ~~ B ) |
5 |
4
|
expr |
|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ A ~<_ B ) -> ( B ~< ~P A -> A ~~ B ) ) |
6 |
5
|
adantrr |
|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( B ~< ~P A -> A ~~ B ) ) |
7 |
6
|
orim1d |
|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( ( B ~< ~P A \/ B ~~ ~P A ) -> ( A ~~ B \/ B ~~ ~P A ) ) ) |
8 |
3 7
|
mpd |
|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( A ~~ B \/ B ~~ ~P A ) ) |