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Theorem elqsn0 8726
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 2737 . 2 (𝐴 / 𝑅) = (𝐴 / 𝑅)
2 neeq1 3007 . 2 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ≠ ∅ ↔ 𝐵 ≠ ∅))
3 eleq2 2827 . . . 4 (dom 𝑅 = 𝐴 → (𝑥 ∈ dom 𝑅𝑥𝐴))
43biimpar 479 . . 3 ((dom 𝑅 = 𝐴𝑥𝐴) → 𝑥 ∈ dom 𝑅)
5 ecdmn0 8696 . . 3 (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅)
64, 5sylib 217 . 2 ((dom 𝑅 = 𝐴𝑥𝐴) → [𝑥]𝑅 ≠ ∅)
71, 2, 6ectocld 8724 1 ((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2944  c0 4283  dom cdm 5634  [cec 8647   / cqs 8648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8651  df-qs 8655
This theorem is referenced by:  ecelqsdm  8727  0nsr  11016  sylow1lem3  19383  vitalilem5  24979  prtlem400  37335  prjspnn0  40963
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