Theorem ndisj2 39218

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 4634	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
3	2	notbii 310	. 2 ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
4		rexnal 2995	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
5		rexnal 2995	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅))
6		ioran 511	. . . . . 6 ⊢ (¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵 ∩ 𝐶) = ∅))
7		df-ne 2795	. . . . . . 7 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
8		df-ne 2795	. . . . . . 7 ⊢ ((𝐵 ∩ 𝐶) ≠ ∅ ↔ ¬ (𝐵 ∩ 𝐶) = ∅)
9	7, 8	anbi12i 733	. . . . . 6 ⊢ ((𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅) ↔ (¬ 𝑥 = 𝑦 ∧ ¬ (𝐵 ∩ 𝐶) = ∅))
10	6, 9	bitr4i 267	. . . . 5 ⊢ (¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
11	10	rexbii 3041	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
12	5, 11	bitr3i 266	. . 3 ⊢ (¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
13	12	rexbii 3041	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐵 ∩ 𝐶) = ∅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
14	3, 4, 13	3bitr2i 288	1 ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ∩ cin 3573  ∅c0 3915  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rmo 2920  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-disj 4621

This theorem is referenced by:  disjrnmpt2  39375