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Theorem n0el3 36325
 Description: Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 27-May-2021.)
Assertion
Ref Expression
n0el3 (¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)

Proof of Theorem n0el3
StepHypRef Expression
1 n0el2 36030 . . . . 5 (¬ ∅ ∈ 𝐴 ↔ dom ( E ↾ 𝐴) = 𝐴)
21biimpi 219 . . . 4 (¬ ∅ ∈ 𝐴 → dom ( E ↾ 𝐴) = 𝐴)
32qseq1d 35987 . . 3 (¬ ∅ ∈ 𝐴 → (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = (𝐴 / ( E ↾ 𝐴)))
4 qsresid 36022 . . . 4 (𝐴 / ( E ↾ 𝐴)) = (𝐴 / E )
5 qsid 8373 . . . 4 (𝐴 / E ) = 𝐴
64, 5eqtri 2781 . . 3 (𝐴 / ( E ↾ 𝐴)) = 𝐴
73, 6eqtrdi 2809 . 2 (¬ ∅ ∈ 𝐴 → (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
8 n0eldmqseq 36324 . 2 ((dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴 → ¬ ∅ ∈ 𝐴)
97, 8impbii 212 1 (¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∅c0 4225   E cep 5434  ◡ccnv 5523  dom cdm 5524   ↾ cres 5526   / cqs 8298 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-eprel 5435  df-xp 5530  df-rel 5531  df-cnv 5532  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ec 8301  df-qs 8305 This theorem is referenced by:  cnvepresdmqss  36326  cnvepresdmqs  36327
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