Theorem clel3g 2898

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 2260	. . 3 |- ( x = B -> ( A e. x <-> A e. B ) )
2	1	ceqsexgv 2893	. 2 |- ( B e. V -> ( E. x ( x = B /\ A e. x ) <-> A e. B ) )
3	2	bicomd 141	1 |- ( B e. V -> ( A e. B <-> E. x ( x = B /\ A e. x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  clel3  2899  dfiun2g  3948