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Theorem n0eldmqs 38008
Description: The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 2-Mar-2018.)
Assertion
Ref Expression
n0eldmqs ¬ ∅ ∈ (dom 𝑅 / 𝑅)

Proof of Theorem n0eldmqs
StepHypRef Expression
1 ssid 3996 . 2 dom 𝑅 ⊆ dom 𝑅
2 n0elqs 37685 . 2 (¬ ∅ ∈ (dom 𝑅 / 𝑅) ↔ dom 𝑅 ⊆ dom 𝑅)
31, 2mpbir 230 1 ¬ ∅ ∈ (dom 𝑅 / 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2098  wss 3940  c0 4314  dom cdm 5666   / cqs 8698
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 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
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 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ec 8701  df-qs 8705
This theorem is referenced by:  n0eldmqseq  38009
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