Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjiun2 | Structured version Visualization version GIF version |
Description: In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
disjiun2.1 | ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) |
disjiun2.2 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
disjiun2.3 | ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶)) |
disjiun2.4 | ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐸) |
Ref | Expression |
---|---|
disjiun2 | ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjiun2.3 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶)) | |
2 | disjiun2.4 | . . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐸) | |
3 | 2 | iunxsng 5014 | . . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸) |
5 | 4 | ineq2d 4191 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸)) |
6 | disjiun2.1 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) | |
7 | disjiun2.2 | . . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
8 | eldifi 4105 | . . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ 𝐴) | |
9 | snssi 4743 | . . . 4 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴) | |
10 | 1, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → {𝐷} ⊆ 𝐴) |
11 | 1 | eldifbd 3951 | . . . 4 ⊢ (𝜑 → ¬ 𝐷 ∈ 𝐶) |
12 | disjsn 4649 | . . . 4 ⊢ ((𝐶 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷 ∈ 𝐶) | |
13 | 11, 12 | sylibr 236 | . . 3 ⊢ (𝜑 → (𝐶 ∩ {𝐷}) = ∅) |
14 | disjiun 5055 | . . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴 ∧ (𝐶 ∩ {𝐷}) = ∅)) → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅) | |
15 | 6, 7, 10, 13, 14 | syl13anc 1368 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅) |
16 | 5, 15 | eqtr3d 2860 | 1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 ∪ ciun 4921 Disj wdisj 5033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-sn 4570 df-iun 4923 df-disj 5034 |
This theorem is referenced by: caratheodorylem1 42815 |
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