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Theorem n0elqs2 38328
Description: Two ways of expressing that the empty set is not an element of a quotient set. (Contributed by Peter Mazsa, 25-Jul-2021.)
Assertion
Ref Expression
n0elqs2 (¬ ∅ ∈ (𝐴 / 𝑅) ↔ dom (𝑅𝐴) = 𝐴)

Proof of Theorem n0elqs2
StepHypRef Expression
1 n0elqs 38327 . 2 (¬ ∅ ∈ (𝐴 / 𝑅) ↔ 𝐴 ⊆ dom 𝑅)
2 ssdmres 6031 . 2 (𝐴 ⊆ dom 𝑅 ↔ dom (𝑅𝐴) = 𝐴)
31, 2bitri 275 1 (¬ ∅ ∈ (𝐴 / 𝑅) ↔ dom (𝑅𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1540  wcel 2108  wss 3951  c0 4333  dom cdm 5685  cres 5687   / 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: (None)
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