MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elqs Unicode version

Theorem elqs 6728
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 6727 . 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 176    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   [cec 6674   /.cqs 6675
This theorem is referenced by:  qsss  6736  qsid  6741  erovlem  6770  sylow2blem3  14949  divsabl  15173  cldsubg  17809  divstgplem  17819  prter2  26852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803  df-qs 6682
  Copyright terms: Public domain W3C validator