Metamath Proof Explorer

Theorem elpwb

Description: Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021)

Ref Expression
Assertion elpwb 
|- ( A e. ~P B <-> ( A e. _V /\ A C_ B ) )

Proof

Step Hyp Ref Expression
1 elex 
 |-  ( A e. ~P B -> A e. _V )
2 elpwg 
 |-  ( A e. _V -> ( A e. ~P B <-> A C_ B ) )
3 1 2 biadanii 
 |-  ( A e. ~P B <-> ( A e. _V /\ A C_ B ) )