Description: Membership in a power class. Theorem 86 of Suppes p. 47. See also elpw2g . (Contributed by NM, 6-Aug-2000) (Proof shortened by BJ, 31-Dec-2023)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | elpwg | |- ( A e. V -> ( A e. ~P B <-> A C_ B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sseq1 | |- ( x = A -> ( x C_ B <-> A C_ B ) ) |
|
| 2 | df-pw | |- ~P B = { x | x C_ B } |
|
| 3 | 1 2 | elab2g | |- ( A e. V -> ( A e. ~P B <-> A C_ B ) ) |