Theorem risset 2570

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 1657	. 2 |- ( E. x ( x e. B /\ x = A ) <-> E. x ( x = A /\ x e. B ) )
2		df-rex 2526	. 2 |- ( E. x e. B x = A <-> E. x ( x e. B /\ x = A ) )
3		df-clel 2228	. 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 1398   E.wex 1541   e. wcel 2203   E.wrex 2521

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-clel 2228  df-rex 2526

This theorem is referenced by:  clel5  2954  reueq  3016  reuind  3022  0el  3531  iunid  4047  sucel  4531  reusv3  4581  fvmptt  5769  releldm2  6379  qsid  6834  rerecclap  9004  nndiv  9278  zq  9958  4fvwrd4  10474  conjnmzb  13997  bj-bdcel  16607