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Mirrors > Home > MPE Home > Th. List > disjord | Structured version Visualization version GIF version |
Description: Conditions for a collection of sets 𝐴(𝑎) for 𝑎 ∈ 𝑉 to be disjoint. (Contributed by AV, 9-Jan-2022.) |
Ref | Expression |
---|---|
disjord.1 | ⊢ (𝑎 = 𝑏 → 𝐴 = 𝐵) |
disjord.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑎 = 𝑏) |
Ref | Expression |
---|---|
disjord | ⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 861 | . . . . . 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 1113 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 → 𝑎 = 𝑏)) |
5 | 4 | con3d 155 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑎 = 𝑏 → ¬ 𝑥 ∈ 𝐵)) |
6 | 5 | impancom 452 | . . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) |
7 | 6 | ralrimiv 3178 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
8 | disj 4395 | . . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | |
9 | 7, 8 | sylibr 235 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝐴 ∩ 𝐵) = ∅) |
10 | 9 | olcd 870 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)) |
11 | 10 | expcom 414 | . . . . 5 ⊢ (¬ 𝑎 = 𝑏 → (𝜑 → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅))) |
12 | 2, 11 | pm2.61i 183 | . . . 4 ⊢ (𝜑 → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)) |
13 | 12 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)) |
14 | 13 | ralrimivva 3188 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)) |
15 | disjord.1 | . . 3 ⊢ (𝑎 = 𝑏 → 𝐴 = 𝐵) | |
16 | 15 | disjor 5037 | . 2 ⊢ (Disj 𝑎 ∈ 𝑉 𝐴 ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)) |
17 | 14, 16 | sylibr 235 | 1 ⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∩ cin 3932 ∅c0 4288 Disj wdisj 5022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rmo 3143 df-v 3494 df-dif 3936 df-in 3940 df-nul 4289 df-disj 5023 |
This theorem is referenced by: 2wspdisj 27668 |
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