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Theorem elqs 7968
 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 7967 . 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 196   = wceq 1632   ∈ wcel 2139  ∃wrex 3051  Vcvv 3340  [cec 7911   / cqs 7912 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-v 3342  df-qs 7919 This theorem is referenced by:  qsss  7977  qsid  7982  erovlem  8012  sylow2blem3  18257  qusabl  18488  cldsubg  22135  qustgplem  22145  n0elqs  34440  prter2  34688
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