MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elqsn0 Structured version   Visualization version   GIF version

Theorem elqsn0 8826
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 3003 . 2 ([𝑥]𝑅 = 𝐵 → ([𝑥]𝑅 ≠ ∅ ↔ 𝐵 ≠ ∅))
3 eleq2 2830 . . . 4 (dom 𝑅 = 𝐴 → (𝑥 ∈ dom 𝑅𝑥𝐴))
43biimpar 477 . . 3 ((dom 𝑅 = 𝐴𝑥𝐴) → 𝑥 ∈ dom 𝑅)
5 ecdmn0 8794 . . 3 (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅)
64, 5sylib 218 . 2 ((dom 𝑅 = 𝐴𝑥𝐴) → [𝑥]𝑅 ≠ ∅)
71, 2, 6ectocld 8824 1 ((dom 𝑅 = 𝐴𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  c0 4333  dom cdm 5685  [cec 8743   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751
This theorem is referenced by:  ecelqsdm  8827  0nsr  11119  sylow1lem3  19618  vitalilem5  25647  prtlem400  38871  prjspnn0  42632
  Copyright terms: Public domain W3C validator