Step |
Hyp |
Ref |
Expression |
1 |
|
gchdju1 |
|- ( ( A e. GCH /\ -. A e. Fin ) -> ( A |_| 1o ) ~~ A ) |
2 |
1
|
ensymd |
|- ( ( A e. GCH /\ -. A e. Fin ) -> A ~~ ( A |_| 1o ) ) |
3 |
|
isfin4-2 |
|- ( A e. GCH -> ( A e. Fin4 <-> -. _om ~<_ A ) ) |
4 |
3
|
adantr |
|- ( ( A e. GCH /\ -. A e. Fin ) -> ( A e. Fin4 <-> -. _om ~<_ A ) ) |
5 |
|
isfin4p1 |
|- ( A e. Fin4 <-> A ~< ( A |_| 1o ) ) |
6 |
|
sdomnen |
|- ( A ~< ( A |_| 1o ) -> -. A ~~ ( A |_| 1o ) ) |
7 |
5 6
|
sylbi |
|- ( A e. Fin4 -> -. A ~~ ( A |_| 1o ) ) |
8 |
4 7
|
syl6bir |
|- ( ( A e. GCH /\ -. A e. Fin ) -> ( -. _om ~<_ A -> -. A ~~ ( A |_| 1o ) ) ) |
9 |
2 8
|
mt4d |
|- ( ( A e. GCH /\ -. A e. Fin ) -> _om ~<_ A ) |