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Theorem elqs 6959
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 6958 . 2  |-  ( B  e.  _V  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
) )
31, 2ax-mp 8 1  |-  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958   [cec 6905   /.cqs 6906
This theorem is referenced by:  qsss  6967  qsid  6972  erovlem  7002  sylow2blem3  15258  divsabl  15482  cldsubg  18142  divstgplem  18152  prter2  26732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-v 2960  df-qs 6913
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