Description: The universe contains every set. (Contributed by NM, 11-Sep-2006) Remove dependency on ax-10 , ax-11 , ax-13 . (Revised by BJ, 10-Aug-2022)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | eqv | |- ( A = _V <-> A. x x e. A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfcleq | |- ( A = _V <-> A. x ( x e. A <-> x e. _V ) ) |
|
| 2 | vex | |- x e. _V |
|
| 3 | 2 | tbt | |- ( x e. A <-> ( x e. A <-> x e. _V ) ) |
| 4 | 3 | albii | |- ( A. x x e. A <-> A. x ( x e. A <-> x e. _V ) ) |
| 5 | 1 4 | bitr4i | |- ( A = _V <-> A. x x e. A ) |