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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjuniel | Structured version Visualization version GIF version |
Description: A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.) |
Ref | Expression |
---|---|
disjuniel.1 | ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥) |
disjuniel.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
disjuniel.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵)) |
Ref | Expression |
---|---|
disjuniel | ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniiun 4509 | . . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥 | |
2 | 1 | ineq1i 3772 | . 2 ⊢ (∪ 𝐵 ∩ 𝐶) = (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶) |
3 | disjuniel.1 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥) | |
4 | id 22 | . . 3 ⊢ (𝑥 = 𝐶 → 𝑥 = 𝐶) | |
5 | disjuniel.2 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
6 | disjuniel.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵)) | |
7 | 3, 4, 5, 6 | disjiunel 28791 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶) = ∅) |
8 | 2, 7 | syl5eq 2656 | 1 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 ∪ ciun 4455 Disj wdisj 4553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-uni 4373 df-iun 4457 df-disj 4554 |
This theorem is referenced by: carsgclctunlem1 29706 |
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