Theorem ndisj2 45893

Description: A non-disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
ndisj2.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

Ref	Expression
ndisj2	⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem ndisj2

Step	Hyp	Ref	Expression
1		ndisj2.1	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
2	1	disjor 5089	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
3	2	notbii 323	. 2 ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
4		rexnal 3116	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
5		rexnal 3116	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
6		ioran 999	. . . . . 6 ⊢ (¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵 ∩ 𝐶) = ∅))
7		df-ne 2958	. . . . . . 7 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
8		df-ne 2958	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) ≠ ∅ ↔ ¬ (𝐵 ∩ 𝐶) = ∅)
9	7, 8	anbi12i 640	. . . . . 6 ⊢ ((𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵 ∩ 𝐶) = ∅))
10	6, 9	bitr4i 281	. . . . 5 ⊢ (¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
11	10	rexbii 3111	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
12	5, 11	bitr3i 280	. . 3 ⊢ (¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
13	12	rexbii 3111	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
14	3, 4, 13	3bitr2i 302	1 ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ≠ wne 2957  ∀wral 3078  ∃wrex 3088   ∩ cin 3901  ∅c0 4282  Disj wdisj 5074

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-v 3455  df-dif 3905  df-in 3909  df-nul 4283  df-disj 5075

This theorem is used by:  disjrnmpt2  46028