Metamath Proof Explorer

Theorem clel4

Description: Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993)

Ref Expression
Hypothesis clel4.1 
|- B e. _V
Assertion clel4 
|- ( A e. B <-> A. x ( x = B -> A e. x ) )

Proof

Step Hyp Ref Expression
1 clel4.1 
 |-  B e. _V
2 clel4g 
 |-  ( B e. _V -> ( A e. B <-> A. x ( x = B -> A e. x ) ) )
3 1 2 ax-mp 
 |-  ( A e. B <-> A. x ( x = B -> A e. x ) )