Theorem risset 2498

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 1601	. 2 |- ( E. x ( x e. B /\ x = A ) <-> E. x ( x = A /\ x e. B ) )
2		df-rex 2454	. 2 |- ( E. x e. B x = A <-> E. x ( x e. B /\ x = A ) )
3		df-clel 2166	. 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 1348   E.wex 1485   e. wcel 2141   E.wrex 2449

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-clel 2166  df-rex 2454

This theorem is referenced by:  clel5  2867  reueq  2929  reuind  2935  0el  3437  iunid  3928  sucel  4395  reusv3  4445  fvmptt  5587  releldm2  6164  qsid  6578  rerecclap  8647  nndiv  8919  zq  9585  4fvwrd4  10096  bj-bdcel  13872