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Theorem elqs 8809
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 8808 . 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 206   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  [cec 8743   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-qs 8751
This theorem is referenced by:  qsss  8818  qsid  8823  erovlem  8853  sylow2blem3  19640  qusabl  19883  cldsubg  24119  qustgplem  24129  qsxpid  33390  n0elqs  38327  prter2  38882
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