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Theorem eldisjn0el 38334
Description: Special case of disjdmqseq 38333 (perhaps this is the closest theorem to the former prter2 38409). (Contributed by Peter Mazsa, 26-Sep-2021.)
Assertion
Ref Expression
eldisjn0el ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴))

Proof of Theorem eldisjn0el
StepHypRef Expression
1 disjdmqseq 38333 . 2 ( Disj ( E ↾ 𝐴) → ((dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴 ↔ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴))
2 df-eldisj 38235 . 2 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
3 n0el3 38179 . . 3 (¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
4 dmqs1cosscnvepreseq 38190 . . . 4 ((dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
54bicomi 223 . . 3 (( 𝐴 /𝐴) = 𝐴 ↔ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴)
63, 5bibi12i 338 . 2 ((¬ ∅ ∈ 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴) ↔ ((dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴 ↔ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴))
71, 2, 63imtr4i 291 1 ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1533  wcel 2098  c0 4318   cuni 4903   E cep 5575  ccnv 5671  dom cdm 5672  cres 5674   / cqs 8722  ccoss 37705  ccoels 37706   Disj wdisjALTV 37739   ElDisj weldisj 37741
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8725  df-qs 8729  df-coss 37939  df-coels 37940  df-cnvrefrel 38055  df-disjALTV 38233  df-eldisj 38235
This theorem is referenced by:  mainer  38362
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