Theorem risset 2536

Description: Two ways to say " A belongs to B". (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
risset	|- ( A e. B <-> E. x e. B x = A )

Distinct variable groups:   x, A   x, B

Proof of Theorem risset

Step	Hyp	Ref	Expression
1		exancom 1632	. 2 |- ( E. x ( x e. B /\ x = A ) <-> E. x ( x = A /\ x e. B ) )
2		df-rex 2492	. 2 |- ( E. x e. B x = A <-> E. x ( x e. B /\ x = A ) )
3		df-clel 2203	. 2 |- ( A e. B <-> E. x ( x = A /\ x e. B ) )
4	1, 2, 3	3bitr4ri 213	1 |- ( A e. B <-> E. x e. B x = A )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   E.wrex 2487

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-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-clel 2203  df-rex 2492

This theorem is referenced by:  clel5  2917  reueq  2979  reuind  2985  0el  3491  iunid  3997  sucel  4475  reusv3  4525  fvmptt  5694  releldm2  6294  qsid  6710  rerecclap  8838  nndiv  9112  zq  9782  4fvwrd4  10297  conjnmzb  13731  bj-bdcel  15972