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Theorem elqsi 6449
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 6447 . 2  |-  ( B  e.  ( A /. R )  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
) )
21ibi 175 1  |-  ( B  e.  ( A /. R )  ->  E. x  e.  A  B  =  [ x ] R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   E.wrex 2394   [cec 6395   /.cqs 6396
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-qs 6403
This theorem is referenced by:  ectocld  6463  ecoptocl  6484  eroveu  6488  dmaddpqlem  7153  nqpi  7154  nq0nn  7218
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