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Mirrors > Home > MPE Home > Th. List > disjiunb | Structured version Visualization version GIF version |
Description: Two ways to say that a collection of index unions 𝐶(𝑖, 𝑥) for 𝑖 ∈ 𝐴 and 𝑥 ∈ 𝐵 is disjoint. (Contributed by AV, 9-Jan-2022.) |
Ref | Expression |
---|---|
disjiunb.1 | ⊢ (𝑖 = 𝑗 → 𝐵 = 𝐷) |
disjiunb.2 | ⊢ (𝑖 = 𝑗 → 𝐶 = 𝐸) |
Ref | Expression |
---|---|
disjiunb | ⊢ (Disj 𝑖 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (∪ 𝑥 ∈ 𝐵 𝐶 ∩ ∪ 𝑥 ∈ 𝐷 𝐸) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjiunb.1 | . . 3 ⊢ (𝑖 = 𝑗 → 𝐵 = 𝐷) | |
2 | disjiunb.2 | . . 3 ⊢ (𝑖 = 𝑗 → 𝐶 = 𝐸) | |
3 | 1, 2 | iuneq12d 5026 | . 2 ⊢ (𝑖 = 𝑗 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐷 𝐸) |
4 | 3 | disjor 5130 | 1 ⊢ (Disj 𝑖 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (∪ 𝑥 ∈ 𝐵 𝐶 ∩ ∪ 𝑥 ∈ 𝐷 𝐸) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1537 ∀wral 3059 ∩ cin 3962 ∅c0 4339 ∪ ciun 4996 Disj wdisj 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rmo 3378 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-nul 4340 df-iun 4998 df-disj 5116 |
This theorem is referenced by: disjiund 5139 otiunsndisj 5530 s3iunsndisj 15004 |
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