Step |
Hyp |
Ref |
Expression |
1 |
|
domeng |
|- ( A e. Fin4 -> ( B ~<_ A <-> E. x ( B ~~ x /\ x C_ A ) ) ) |
2 |
1
|
biimpa |
|- ( ( A e. Fin4 /\ B ~<_ A ) -> E. x ( B ~~ x /\ x C_ A ) ) |
3 |
|
ensym |
|- ( B ~~ x -> x ~~ B ) |
4 |
3
|
ad2antrl |
|- ( ( ( A e. Fin4 /\ B ~<_ A ) /\ ( B ~~ x /\ x C_ A ) ) -> x ~~ B ) |
5 |
|
ssfin4 |
|- ( ( A e. Fin4 /\ x C_ A ) -> x e. Fin4 ) |
6 |
5
|
ad2ant2rl |
|- ( ( ( A e. Fin4 /\ B ~<_ A ) /\ ( B ~~ x /\ x C_ A ) ) -> x e. Fin4 ) |
7 |
|
fin4en1 |
|- ( x ~~ B -> ( x e. Fin4 -> B e. Fin4 ) ) |
8 |
4 6 7
|
sylc |
|- ( ( ( A e. Fin4 /\ B ~<_ A ) /\ ( B ~~ x /\ x C_ A ) ) -> B e. Fin4 ) |
9 |
2 8
|
exlimddv |
|- ( ( A e. Fin4 /\ B ~<_ A ) -> B e. Fin4 ) |