Theorem eldisjn0el 40032

Description: Special case of disjdmqseq 40031 (perhaps this is the closest theorem to the former prter2 40129). (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 40031	. 2 ⊢ ( Disj (◡ E ↾ 𝐴) → ((dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴 ↔ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴))
2		df-eldisj 39915	. 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
3		n0el3 39859	. . 3 ⊢ (¬ ∅ ∈ 𝐴 ↔ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)
4		dmqs1cosscnvepreseq 39870	. . . 4 ⊢ ((dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)
5	4	bicomi 227	. . 3 ⊢ ((∪ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴)
6	3, 5	bibi12i 342	. 2 ⊢ ((¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴) ↔ ((dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴 ↔ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴))
7	1, 2, 6	3imtr4i 295	1 ⊢ ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∅c0 4281  ∪ cuni 4870   E cep 5554  ^◡ccnv 5654  dom cdm 5655   ↾ cres 5657   / cqs 8741   ≀ ccoss 39306   ∼ ccoels 39307   Disj wdisjALTV 39342   ElDisj weldisj 39344

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8744  df-qs 8748  df-coss 39624  df-coels 39625  df-cnvrefrel 39730  df-disjALTV 39913  df-eldisj 39915

This theorem is used by:  mainer  40071