Description: No set equals its power set. The sethood antecedent is necessary; compare pwv . (Contributed by NM, 17-Nov-2008) (Proof shortened by Mario Carneiro, 23-Dec-2016)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | pwne | |- ( A e. V -> ~P A =/= A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pwnss | |- ( A e. V -> -. ~P A C_ A ) |
|
| 2 | eqimss | |- ( ~P A = A -> ~P A C_ A ) |
|
| 3 | 2 | necon3bi | |- ( -. ~P A C_ A -> ~P A =/= A ) |
| 4 | 1 3 | syl | |- ( A e. V -> ~P A =/= A ) |