Description: The size of an infinite set is not a nonnegative integer. (Contributed by Alexander van der Vekens, 21-Dec-2017) (Proof shortened by Alexander van der Vekens, 18-Jan-2018)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | hashnfinnn0 | |- ( ( A e. V /\ -. A e. Fin ) -> ( # ` A ) e/ NN0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nnel | |- ( -. ( # ` A ) e/ NN0 <-> ( # ` A ) e. NN0 ) |
|
| 2 | hashclb | |- ( A e. V -> ( A e. Fin <-> ( # ` A ) e. NN0 ) ) |
|
| 3 | 2 | biimprd | |- ( A e. V -> ( ( # ` A ) e. NN0 -> A e. Fin ) ) |
| 4 | 1 3 | syl5bi | |- ( A e. V -> ( -. ( # ` A ) e/ NN0 -> A e. Fin ) ) |
| 5 | 4 | con1d | |- ( A e. V -> ( -. A e. Fin -> ( # ` A ) e/ NN0 ) ) |
| 6 | 5 | imp | |- ( ( A e. V /\ -. A e. Fin ) -> ( # ` A ) e/ NN0 ) |