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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 5113 | . . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸) |
5 | 4 | ineq2d 4241 | . 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 4154 | . . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ 𝐴) | |
9 | snssi 4833 | . . . 4 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴) | |
10 | 1, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → {𝐷} ⊆ 𝐴) |
11 | 1 | eldifbd 3989 | . . . 4 ⊢ (𝜑 → ¬ 𝐷 ∈ 𝐶) |
12 | disjsn 4736 | . . . 4 ⊢ ((𝐶 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷 ∈ 𝐶) | |
13 | 11, 12 | sylibr 234 | . . 3 ⊢ (𝜑 → (𝐶 ∩ {𝐷}) = ∅) |
14 | disjiun 5154 | . . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴 ∧ (𝐶 ∩ {𝐷}) = ∅)) → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅) | |
15 | 6, 7, 10, 13, 14 | syl13anc 1372 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅) |
16 | 5, 15 | eqtr3d 2782 | 1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2108 ∖ cdif 3973 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 {csn 4648 ∪ ciun 5015 Disj wdisj 5133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-clab 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-rab 3444 df-v 3490 df-dif 3979 df-in 3983 df-ss 3993 df-nul 4353 df-sn 4649 df-iun 5017 df-disj 5134 |
This theorem is referenced by: caratheodorylem1 46447 |
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