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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 4974 | . . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥 | |
2 | 1 | ineq1i 4184 | . 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 30340 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶) = ∅) |
8 | 2, 7 | syl5eq 2868 | 1 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4831 ∪ 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-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-sn 4561 df-uni 4832 df-iun 4913 df-disj 5024 |
This theorem is referenced by: carsgclctunlem1 31570 |
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