Theorem risset 2494

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 1596	. 2 |- ( E. x ( x e. B /\ x = A ) <-> E. x ( x = A /\ x e. B ) )
2		df-rex 2450	. 2 |- ( E. x e. B x = A <-> E. x ( x e. B /\ x = A ) )
3		df-clel 2161	. 2 |- ( A e. B <-> E. x ( x = A /\ x e. B ) )
4	1, 2, 3	3bitr4ri 212	1 |- ( A e. B <-> E. x e. B x = A )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   E.wrex 2445

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-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-clel 2161  df-rex 2450

This theorem is referenced by:  clel5  2863  reueq  2925  reuind  2931  0el  3431  iunid  3921  sucel  4388  reusv3  4438  fvmptt  5577  releldm2  6153  qsid  6566  rerecclap  8626  nndiv  8898  zq  9564  4fvwrd4  10075  bj-bdcel  13719