Metamath Proof Explorer

Theorem elpwgOLD

Description: Obsolete proof of elpwg as of 31-Dec-2023. (Contributed by NM, 6-Aug-2000) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion elpwgOLD 
|- ( A e. V -> ( A e. ~P B <-> A C_ B ) )

Proof

Step Hyp Ref Expression
1 eleq1 
 |-  ( x = A -> ( x e. ~P B <-> A e. ~P B ) )
2 sseq1 
 |-  ( x = A -> ( x C_ B <-> A C_ B ) )
3 velpw 
 |-  ( x e. ~P B <-> x C_ B )
4 1 2 3 vtoclbg 
 |-  ( A e. V -> ( A e. ~P B <-> A C_ B ) )