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Theorem elqsi 6713
Description: Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
elqsi  |-  ( B  e.  ( A /. R )  ->  E. x  e.  A  B  =  [ x ] R
)
Distinct variable groups:    x, A    x, B    x, R

Proof of Theorem elqsi
StepHypRef Expression
1 elqsg 6711 . 2  |-  ( B  e.  ( A /. R )  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
) )
21ibi 232 1  |-  ( B  e.  ( A /. R )  ->  E. x  e.  A  B  =  [ x ] R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544   [cec 6658   /.cqs 6659
This theorem is referenced by:  ectocld  6726  ecoptocl  6748  eroveu  6753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-qs 6666
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