Theorem clel3 2824

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 2823	. 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 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691

This theorem is referenced by:  unipr  3758