Metamath Proof Explorer

Theorem elpwg

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 ) )

Proof

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 ) )