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Theorem eldisjn0el 38807
Description: Special case of disjdmqseq 38806 (perhaps this is the closest theorem to the former prter2 38882). (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 38806 . 2 ( Disj ( E ↾ 𝐴) → ((dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴 ↔ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴))
2 df-eldisj 38708 . 2 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
3 n0el3 38652 . . 3 (¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
4 dmqs1cosscnvepreseq 38663 . . . 4 ((dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
54bicomi 224 . . 3 (( 𝐴 /𝐴) = 𝐴 ↔ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴)
63, 5bibi12i 339 . 2 ((¬ ∅ ∈ 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴) ↔ ((dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴 ↔ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴))
71, 2, 63imtr4i 292 1 ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1540  wcel 2108  c0 4333   cuni 4907   E cep 5583  ccnv 5684  dom cdm 5685  cres 5687   / cqs 8744  ccoss 38182  ccoels 38183   Disj wdisjALTV 38216   ElDisj weldisj 38218
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-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  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-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751  df-coss 38412  df-coels 38413  df-cnvrefrel 38528  df-disjALTV 38706  df-eldisj 38708
This theorem is referenced by:  mainer  38835
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