Theorem risset 2485

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 2441	. 2 |- ( E. x e. B x = A <-> E. x ( x e. B /\ x = A ) )
3		df-clel 2153	. 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 1335   E.wex 1472   e. wcel 2128   E.wrex 2436

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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-clel 2153  df-rex 2441

This theorem is referenced by:  clel5  2849  reueq  2911  reuind  2917  0el  3416  iunid  3904  sucel  4369  reusv3  4418  fvmptt  5556  releldm2  6127  qsid  6538  rerecclap  8586  nndiv  8857  zq  9517  4fvwrd4  10021  bj-bdcel  13371