Theorem disjord 5155

Description: Conditions for a collection of sets 𝐴(𝑎) for 𝑎 ∈ 𝑉 to be disjoint. (Contributed by AV, 9-Jan-2022.)

Hypotheses

Ref	Expression
disjord.1	⊢ (𝑎 = 𝑏 → 𝐴 = 𝐵)
disjord.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑎 = 𝑏)

Assertion

Ref	Expression
disjord	⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 𝐴)

Distinct variable groups:   𝐴,𝑏,𝑥   𝐵,𝑎,𝑥   𝑉,𝑎,𝑏,𝑥   𝜑,𝑎,𝑏,𝑥

Allowed substitution hints:   𝐴(𝑎)   𝐵(𝑏)

Proof of Theorem disjord

Step	Hyp	Ref	Expression
1		orc 866	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅))
2	1	a1d 25	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝜑 → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)))
3		disjord.2	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑎 = 𝑏)
4	3	3expia 1121	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 → 𝑎 = 𝑏))
5	4	con3d 152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑎 = 𝑏 → ¬ 𝑥 ∈ 𝐵))
6	5	impancom 451	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
7	6	ralrimiv 3151	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
8		disj 4473	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
9	7, 8	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝐴 ∩ 𝐵) = ∅)
10	9	olcd 873	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅))
11	10	expcom 413	. . . . 5 ⊢ (¬ 𝑎 = 𝑏 → (𝜑 → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)))
12	2, 11	pm2.61i 182	. . . 4 ⊢ (𝜑 → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅))
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅))
14	13	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅))
15		disjord.1	. . 3 ⊢ (𝑎 = 𝑏 → 𝐴 = 𝐵)
16	15	disjor 5148	. 2 ⊢ (Disj 𝑎 ∈ 𝑉 𝐴 ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅))
17	14, 16	sylibr 234	1 ⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∩ cin 3975  ∅c0 4352  Disj wdisj 5133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rmo 3388  df-v 3490  df-dif 3979  df-in 3983  df-nul 4353  df-disj 5134

This theorem is referenced by:  2wspdisj  29995