Theorem risset 2466

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 1588	. 2 |- ( E. x ( x e. B /\ x = A ) <-> E. x ( x = A /\ x e. B ) )
2		df-rex 2423	. 2 |- ( E. x e. B x = A <-> E. x ( x e. B /\ x = A ) )
3		df-clel 2136	. 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 1332   E.wex 1469   e. wcel 1481   E.wrex 2418

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-clel 2136  df-rex 2423

This theorem is referenced by:  reueq  2887  reuind  2893  0el  3390  iunid  3876  sucel  4340  reusv3  4389  fvmptt  5520  releldm2  6091  qsid  6502  rerecclap  8514  nndiv  8785  zq  9445  4fvwrd4  9948  bj-bdcel  13206