Theorem risset 2572

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 2528	. 2 |- ( E. x e. B x = A <-> E. x ( x e. B /\ x = A ) )
3		df-clel 2230	. 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 2205   E.wrex 2523

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 2230  df-rex 2528

This theorem is referenced by:  clel5  2956  reueq  3018  reuind  3024  0el  3533  iunid  4049  sucel  4533  reusv3  4583  fvmptt  5771  releldm2  6381  qsid  6836  rerecclap  9006  nndiv  9280  zq  9961  4fvwrd4  10478  conjnmzb  14014  bj-bdcel  16624