Theorem clel3 2941

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 2940	. 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:   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804

This theorem is referenced by:  unipr  3907