Description: The size of a set is greater than zero if and only if the set contains at least one element. (Contributed by Alexander van der Vekens, 18-Jan-2018)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | hashgt0elexb | |- ( V e. W -> ( 0 < ( # ` V ) <-> E. x x e. V ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hashgt0elex | |- ( ( V e. W /\ 0 < ( # ` V ) ) -> E. x x e. V ) |
|
| 2 | n0 | |- ( V =/= (/) <-> E. x x e. V ) |
|
| 3 | hashgt0 | |- ( ( V e. W /\ V =/= (/) ) -> 0 < ( # ` V ) ) |
|
| 4 | 2 3 | sylan2br | |- ( ( V e. W /\ E. x x e. V ) -> 0 < ( # ` V ) ) |
| 5 | 1 4 | impbida | |- ( V e. W -> ( 0 < ( # ` V ) <-> E. x x e. V ) ) |