Step |
Hyp |
Ref |
Expression |
1 |
|
hashcl |
|- ( A e. Fin -> ( # ` A ) e. NN0 ) |
2 |
|
nn0re |
|- ( ( # ` A ) e. NN0 -> ( # ` A ) e. RR ) |
3 |
|
pnfnre |
|- +oo e/ RR |
4 |
3
|
neli |
|- -. +oo e. RR |
5 |
|
hashinf |
|- ( ( A e. V /\ -. A e. Fin ) -> ( # ` A ) = +oo ) |
6 |
5
|
eleq1d |
|- ( ( A e. V /\ -. A e. Fin ) -> ( ( # ` A ) e. RR <-> +oo e. RR ) ) |
7 |
4 6
|
mtbiri |
|- ( ( A e. V /\ -. A e. Fin ) -> -. ( # ` A ) e. RR ) |
8 |
7
|
ex |
|- ( A e. V -> ( -. A e. Fin -> -. ( # ` A ) e. RR ) ) |
9 |
8
|
con4d |
|- ( A e. V -> ( ( # ` A ) e. RR -> A e. Fin ) ) |
10 |
2 9
|
syl5 |
|- ( A e. V -> ( ( # ` A ) e. NN0 -> A e. Fin ) ) |
11 |
1 10
|
impbid2 |
|- ( A e. V -> ( A e. Fin <-> ( # ` A ) e. NN0 ) ) |