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Theorem elqs 8338
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 8337 . 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 207   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492  [cec 8276   / cqs 8277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-qs 8284
This theorem is referenced by:  qsss  8347  qsid  8352  erovlem  8382  sylow2blem3  18676  qusabl  18914  cldsubg  22646  qustgplem  22656  qsxpid  30854  n0elqs  35464  prter2  35897
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