Theorem risset 2558

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 1654	. 2 |- ( E. x ( x e. B /\ x = A ) <-> E. x ( x = A /\ x e. B ) )
2		df-rex 2514	. 2 |- ( E. x e. B x = A <-> E. x ( x e. B /\ x = A ) )
3		df-clel 2225	. 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 1395   E.wex 1538   e. wcel 2200   E.wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-clel 2225  df-rex 2514

This theorem is referenced by:  clel5  2940  reueq  3002  reuind  3008  0el  3514  iunid  4021  sucel  4501  reusv3  4551  fvmptt  5726  releldm2  6331  qsid  6747  rerecclap  8877  nndiv  9151  zq  9821  4fvwrd4  10336  conjnmzb  13817  bj-bdcel  16200