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Theorem risset 2369
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 1515 . 2  |-  ( E. x ( x  e.  B  /\  x  =  A )  <->  E. x
( x  =  A  /\  x  e.  B
) )
2 df-rex 2329 . 2  |-  ( E. x  e.  B  x  =  A  <->  E. x
( x  e.  B  /\  x  =  A
) )
3 df-clel 2052 . 2  |-  ( A  e.  B  <->  E. x
( x  =  A  /\  x  e.  B
) )
41, 2, 33bitr4ri 206 1  |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-clel 2052  df-rex 2329
This theorem is referenced by:  reueq  2761  reuind  2767  0el  3269  iunid  3740  sucel  4175  reusv3  4220  fvmptt  5290  releldm2  5839  qsid  6202  rerecclap  7781  nndiv  8030  zq  8658  4fvwrd4  9099  bj-bdcel  10344
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