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Theorem n0eldmqseq 36408
Description: The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 3-Nov-2018.)
Assertion
Ref Expression
n0eldmqseq ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)

Proof of Theorem n0eldmqseq
StepHypRef Expression
1 n0eldmqs 36407 . 2 ¬ ∅ ∈ (dom 𝑅 / 𝑅)
2 eleq2 2822 . 2 ((dom 𝑅 / 𝑅) = 𝐴 → (∅ ∈ (dom 𝑅 / 𝑅) ↔ ∅ ∈ 𝐴))
31, 2mtbii 329 1 ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2114  c0 4212  dom cdm 5526   / cqs 8322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-br 5032  df-opab 5094  df-xp 5532  df-cnv 5534  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-ec 8325  df-qs 8329
This theorem is referenced by:  n0el3  36409
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