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Theorem elqs 8558
Description: Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
Hypothesis
Ref Expression
elqs.1 𝐵 ∈ V
Assertion
Ref Expression
elqs (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem elqs
StepHypRef Expression
1 elqs.1 . 2 𝐵 ∈ V
2 elqsg 8557 . 2 (𝐵 ∈ V → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
31, 2ax-mp 5 1 (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1539  wcel 2106  wrex 3065  Vcvv 3432  [cec 8496   / cqs 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-qs 8504
This theorem is referenced by:  qsss  8567  qsid  8572  erovlem  8602  sylow2blem3  19227  qusabl  19466  cldsubg  23262  qustgplem  23272  qsxpid  31558  n0elqs  36461  prter2  36895
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