Theorem clel3g 2860

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 2230	. . 3 |- ( x = B -> ( A e. x <-> A e. B ) )
2	1	ceqsexgv 2855	. 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 1343   E.wex 1480   e. wcel 2136

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  clel3  2861  dfiun2g  3898