Theorem clel3 2679

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 2678	. 2 |- ( B e. _V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) )
3	1, 2	ax-mp 7	1 |- ( A e. B <-> E. x ( x = B /\ A e. x ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  unipr  3594