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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 4939 | . 2 ⊢ (𝑖 = 𝑗 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐷 𝐸) |
4 | 3 | disjor 5038 | 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 208 ∨ wo 843 = wceq 1533 ∀wral 3138 ∩ cin 3934 ∅c0 4290 ∪ ciun 4911 Disj wdisj 5023 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rmo 3146 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 df-iun 4913 df-disj 5024 |
This theorem is referenced by: disjiund 5048 otiunsndisj 5402 s3iunsndisj 14322 |
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