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Theorem eldisjn0el 38189
Description: Special case of disjdmqseq 38188 (perhaps this is the closest theorem to the former prter2 38264). (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 38188 . 2 ( Disj ( E ↾ 𝐴) → ((dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴 ↔ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴))
2 df-eldisj 38090 . 2 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
3 n0el3 38034 . . 3 (¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
4 dmqs1cosscnvepreseq 38045 . . . 4 ((dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
54bicomi 223 . . 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 205   = wceq 1533  wcel 2098  c0 4317   cuni 4902   E cep 5572  ccnv 5668  dom cdm 5669  cres 5671   / cqs 8704  ccoss 37556  ccoels 37557   Disj wdisjALTV 37590   ElDisj weldisj 37592
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
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-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707  df-qs 8711  df-coss 37794  df-coels 37795  df-cnvrefrel 37910  df-disjALTV 38088  df-eldisj 38090
This theorem is referenced by:  mainer  38217
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