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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
StepHypRef 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
) )
41, 2, 33bitr4ri 212 1  |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
Colors of variables: wff set class
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
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