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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 4909 | . 2 ⊢ (𝑖 = 𝑗 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐷 𝐸) |
4 | 3 | disjor 5010 | 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 209 ∨ wo 846 = wceq 1542 ∀wral 3053 ∩ cin 3842 ∅c0 4211 ∪ ciun 4881 Disj wdisj 4995 |
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 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-mo 2540 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rmo 3061 df-v 3400 df-dif 3846 df-in 3850 df-ss 3860 df-nul 4212 df-iun 4883 df-disj 4996 |
This theorem is referenced by: disjiund 5020 otiunsndisj 5377 s3iunsndisj 14419 |
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