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Theorem elqs 6487
Description: Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
Hypothesis
Ref Expression
elqs.1  |-  B  e. 
_V
Assertion
Ref Expression
elqs  |-  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
)
Distinct variable groups:    x, A    x, B    x, R

Proof of Theorem elqs
StepHypRef Expression
1 elqs.1 . 2  |-  B  e. 
_V
2 elqsg 6486 . 2  |-  ( B  e.  _V  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
) )
31, 2ax-mp 5 1  |-  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1332    e. wcel 1481   E.wrex 2418   _Vcvv 2689   [cec 6434   /.cqs 6435
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-qs 6442
This theorem is referenced by:  qsss  6495  qsid  6501  erovlem  6528  nqnq0  7272
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