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Theorem elqsn0 8779
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 2726 . 2 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 neeq1 2997 . 2 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ≠ ∅ ↔ 𝐵 ≠ ∅))
3 eleq2 2816 . . . 4 (dom 𝑅 = 𝐴 → (𝑥 ∈ dom 𝑅𝑥𝐴))
43biimpar 477 . . 3 ((dom 𝑅 = 𝐴𝑥𝐴) → 𝑥 ∈ dom 𝑅)
5 ecdmn0 8749 . . 3 (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅)
64, 5sylib 217 . 2 ((dom 𝑅 = 𝐴𝑥𝐴) → [𝑥]𝑅 ≠ ∅)
71, 2, 6ectocld 8777 1 ((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wne 2934  c0 4317  dom cdm 5669  [cec 8700   / cqs 8701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8704  df-qs 8708
This theorem is referenced by:  ecelqsdm  8780  0nsr  11073  sylow1lem3  19518  vitalilem5  25492  prtlem400  38251  prjspnn0  41923
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