Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of Suppes p. 22. (Contributed by NM, 29-Aug-1993) Avoid ax-11 , ax-12 . (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | eq0 | |- ( A = (/) <-> A. x -. x e. A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfcleq | |- ( A = (/) <-> A. x ( x e. A <-> x e. (/) ) ) |
|
| 2 | noel | |- -. x e. (/) |
|
| 3 | 2 | nbn | |- ( -. x e. A <-> ( x e. A <-> x e. (/) ) ) |
| 4 | 3 | albii | |- ( A. x -. x e. A <-> A. x ( x e. A <-> x e. (/) ) ) |
| 5 | 1 4 | bitr4i | |- ( A = (/) <-> A. x -. x e. A ) |