Theorem clel3g 2846

Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)

Assertion

Ref	Expression
clel3g	|- ( B e. V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem clel3g

Step	Hyp	Ref	Expression
1		eleq2 2221	. . 3 |- ( x = B -> ( A e. x <-> A e. B ) )
2	1	ceqsexgv 2841	. 2 |- ( B e. V -> ( E. x ( x = B /\ A e. x ) <-> A e. B ) )
3	2	bicomd 140	1 |- ( B e. V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   E.wex 1472   e. wcel 2128

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714

This theorem is referenced by:  clel3  2847  dfiun2g  3881