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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 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 1122 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 → 𝑎 = 𝑏)) |
5 | 4 | con3d 155 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑎 = 𝑏 → ¬ 𝑥 ∈ 𝐵)) |
6 | 5 | impancom 455 | . . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) |
7 | 6 | ralrimiv 3096 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
8 | disj 4347 | . . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | |
9 | 7, 8 | sylibr 237 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝐴 ∩ 𝐵) = ∅) |
10 | 9 | olcd 873 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑏) → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)) |
11 | 10 | expcom 417 | . . . . 5 ⊢ (¬ 𝑎 = 𝑏 → (𝜑 → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅))) |
12 | 2, 11 | pm2.61i 185 | . . . 4 ⊢ (𝜑 → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)) |
13 | 12 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)) |
14 | 13 | ralrimivva 3104 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)) |
15 | disjord.1 | . . 3 ⊢ (𝑎 = 𝑏 → 𝐴 = 𝐵) | |
16 | 15 | disjor 5020 | . 2 ⊢ (Disj 𝑎 ∈ 𝑉 𝐴 ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑎 = 𝑏 ∨ (𝐴 ∩ 𝐵) = ∅)) |
17 | 14, 16 | sylibr 237 | 1 ⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ∩ cin 3852 ∅c0 4221 Disj wdisj 5005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-11 2162 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-mo 2541 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rmo 3062 df-v 3402 df-dif 3856 df-in 3860 df-nul 4222 df-disj 5006 |
This theorem is referenced by: 2wspdisj 27912 |
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