Theorem clel3g 2864

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 2234	. . 3 |- ( x = B -> ( A e. x <-> A e. B ) )
2	1	ceqsexgv 2859	. 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 1348   E.wex 1485   e. wcel 2141

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  clel3  2865  dfiun2g  3905