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Theorem elqsi 6950
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 6948 . 2  |-  ( B  e.  ( A /. R )  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
) )
21ibi 233 1  |-  ( B  e.  ( A /. R )  ->  E. x  e.  A  B  =  [ x ] R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   E.wrex 2698   [cec 6895   /.cqs 6896
This theorem is referenced by:  ectocld  6963  ecoptocl  6986  eroveu  6991  pstmxmet  24284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-v 2950  df-qs 6903
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