Theorem eldisjn0el 39346

Description: Special case of disjdmqseq 39345 (perhaps this is the closest theorem to the former prter2 39443). (Contributed by Peter Mazsa, 26-Sep-2021.)

Assertion

Ref	Expression
eldisjn0el	⊢ ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Proof of Theorem eldisjn0el

Step	Hyp	Ref	Expression
1		disjdmqseq 39345	. 2 ⊢ ( Disj (◡ E ↾ 𝐴) → ((dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴 ↔ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴))
2		df-eldisj 39229	. 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
3		n0el3 39173	. . 3 ⊢ (¬ ∅ ∈ 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)
4		dmqs1cosscnvepreseq 39184	. . . 4 ⊢ ((dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)
5	4	bicomi 226	. . 3 ⊢ ((∪ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴)
6	3, 5	bibi12i 341	. 2 ⊢ ((¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴) ↔ ((dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴 ↔ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴))
7	1, 2, 6	3imtr4i 294	1 ⊢ ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1550   ∈ wcel 2132  ∅c0 4276  ∪ cuni 4855   E cep 5535  ^◡ccnv 5635  dom cdm 5636   ↾ cres 5638   / cqs 8661   ≀ ccoss 38620   ∼ ccoels 38621   Disj wdisjALTV 38656   ElDisj weldisj 38658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-id 5531  df-eprel 5536  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-ec 8664  df-qs 8668  df-coss 38938  df-coels 38939  df-cnvrefrel 39044  df-disjALTV 39227  df-eldisj 39229

This theorem is referenced by:  mainer  39385