Theorem ndisj2 43786

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 5129	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
3	2	notbii 320	. 2 ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
4		rexnal 3101	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
5		rexnal 3101	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
6		ioran 983	. . . . . 6 ⊢ (¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵 ∩ 𝐶) = ∅))
7		df-ne 2942	. . . . . . 7 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
8		df-ne 2942	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) ≠ ∅ ↔ ¬ (𝐵 ∩ 𝐶) = ∅)
9	7, 8	anbi12i 628	. . . . . 6 ⊢ ((𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵 ∩ 𝐶) = ∅))
10	6, 9	bitr4i 278	. . . . 5 ⊢ (¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
11	10	rexbii 3095	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
12	5, 11	bitr3i 277	. . 3 ⊢ (¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
13	12	rexbii 3095	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
14	3, 4, 13	3bitr2i 299	1 ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ≠ wne 2941  ∀wral 3062  ∃wrex 3071   ∩ cin 3948  ∅c0 4323  Disj wdisj 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-v 3477  df-dif 3952  df-in 3956  df-nul 4324  df-disj 5115

This theorem is referenced by:  disjrnmpt2  43934