Description: A class is not empty if and only if it has at least one element. Proposition 5.17(1) of TakeutiZaring p. 20. (Contributed by NM, 21-Jun-1993) Avoid ax-11 , ax-12 . (Revised by Gino Giotto, 28-Jun-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | neq0 | |- ( -. A = (/) <-> E. x x e. A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ex | |- ( E. x x e. A <-> -. A. x -. x e. A ) |
|
| 2 | eq0 | |- ( A = (/) <-> A. x -. x e. A ) |
|
| 3 | 1 2 | xchbinxr | |- ( E. x x e. A <-> -. A = (/) ) |
| 4 | 3 | bicomi | |- ( -. A = (/) <-> E. x x e. A ) |