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Theorem elqsn0 8825
Description: A quotient set does not contain the empty set. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
elqsn0 ((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅)

Proof of Theorem elqsn0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . 2 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 neeq1 3001 . 2 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ≠ ∅ ↔ 𝐵 ≠ ∅))
3 eleq2 2828 . . . 4 (dom 𝑅 = 𝐴 → (𝑥 ∈ dom 𝑅𝑥𝐴))
43biimpar 477 . . 3 ((dom 𝑅 = 𝐴𝑥𝐴) → 𝑥 ∈ dom 𝑅)
5 ecdmn0 8793 . . 3 (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅)
64, 5sylib 218 . 2 ((dom 𝑅 = 𝐴𝑥𝐴) → [𝑥]𝑅 ≠ ∅)
71, 2, 6ectocld 8823 1 ((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  c0 4339  dom cdm 5689  [cec 8742   / cqs 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750
This theorem is referenced by:  ecelqsdm  8826  0nsr  11117  sylow1lem3  19633  vitalilem5  25661  prtlem400  38852  prjspnn0  42609
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