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Mathbox for Thierry Arnoux |
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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 5056 | . . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥 | |
2 | 1 | ineq1i 4205 | . 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 32380 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶) = ∅) |
8 | 2, 7 | eqtrid 2780 | 1 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∖ cdif 3942 ∩ cin 3944 ⊆ wss 3945 ∅c0 4319 ∪ cuni 4904 ∪ ciun 4992 Disj wdisj 5108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3058 df-rex 3067 df-rmo 3372 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-sn 4626 df-uni 4905 df-iun 4994 df-disj 5109 |
This theorem is referenced by: carsgclctunlem1 33932 |
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