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Theorem eldisjn0el 38143
Description: Special case of disjdmqseq 38142 (perhaps this is the closest theorem to the former prter2 38218). (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 38142 . 2 ( Disj ( E ↾ 𝐴) → ((dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴 ↔ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴))
2 df-eldisj 38044 . 2 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
3 n0el3 37988 . . 3 (¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
4 dmqs1cosscnvepreseq 37999 . . . 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 1540  wcel 2105  c0 4322   cuni 4908   E cep 5579  ccnv 5675  dom cdm 5676  cres 5678   / cqs 8708  ccoss 37510  ccoels 37511   Disj wdisjALTV 37544   ElDisj weldisj 37546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8711  df-qs 8715  df-coss 37748  df-coels 37749  df-cnvrefrel 37864  df-disjALTV 38042  df-eldisj 38044
This theorem is referenced by:  mainer  38171
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