Theorem risset 2407

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 1545	. 2 |- ( E. x ( x e. B /\ x = A ) <-> E. x ( x = A /\ x e. B ) )
2		df-rex 2366	. 2 |- ( E. x e. B x = A <-> E. x ( x e. B /\ x = A ) )
3		df-clel 2085	. 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 1290   E.wex 1427   e. wcel 1439   E.wrex 2361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-ial 1473

This theorem depends on definitions:  df-bi 116  df-clel 2085  df-rex 2366

This theorem is referenced by:  reueq  2815  reuind  2821  0el  3309  iunid  3791  sucel  4246  reusv3  4295  fvmptt  5407  releldm2  5969  qsid  6371  rerecclap  8258  nndiv  8524  zq  9172  4fvwrd4  9612  bj-bdcel  12001