Theorem clel3 3341

Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
clel3.1	|- B e. _V

Assertion

Ref	Expression
clel3	|- ( A e. B <-> E. x ( x = B /\ A e. x ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem clel3

Step	Hyp	Ref	Expression
1		clel3.1	. 2 |- B e. _V
2		clel3g 3340	. 2 |- ( B e. _V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) )
3	1, 2	ax-mp 5	1 |- ( A e. B <-> E. x ( x = B /\ A e. x ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  unipr  4449  brcup  32046  brcap  32047