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 = y -> ( x C_ B <-> y C_ B ) ) |
|
2 | sseq1 | |- ( y = A -> ( y C_ B <-> A C_ B ) ) |
|
3 | df-pw | |- ~P B = { x | x C_ B } |
|
4 | 1 2 3 | elab2gw | |- ( A e. V -> ( A e. ~P B <-> A C_ B ) ) |