Description: Properties of a finite interval of integers which is nonempty. (Contributed by Jeff Madsen, 17-Jun-2010) (Revised by Mario Carneiro, 28-Apr-2015)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | fzn0 | |- ( ( M ... N ) =/= (/) <-> N e. ( ZZ>= ` M ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0 | |- ( ( M ... N ) =/= (/) <-> E. x x e. ( M ... N ) ) |
|
| 2 | elfzuz2 | |- ( x e. ( M ... N ) -> N e. ( ZZ>= ` M ) ) |
|
| 3 | 2 | exlimiv | |- ( E. x x e. ( M ... N ) -> N e. ( ZZ>= ` M ) ) |
| 4 | 1 3 | sylbi | |- ( ( M ... N ) =/= (/) -> N e. ( ZZ>= ` M ) ) |
| 5 | eluzfz1 | |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) ) |
|
| 6 | 5 | ne0d | |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) =/= (/) ) |
| 7 | 4 6 | impbii | |- ( ( M ... N ) =/= (/) <-> N e. ( ZZ>= ` M ) ) |