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 4538 | . . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ {𝐷}𝐵 = 𝐸) |
5 | 4 | ineq2d 3776 | . 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 3694 | . . . 4 ⊢ (𝐷 ∈ (𝐴 ∖ 𝐶) → 𝐷 ∈ 𝐴) | |
9 | snssi 4280 | . . . 4 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴) | |
10 | 1, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → {𝐷} ⊆ 𝐴) |
11 | 1 | eldifbd 3553 | . . . 4 ⊢ (𝜑 → ¬ 𝐷 ∈ 𝐶) |
12 | disjsn 4192 | . . . 4 ⊢ ((𝐶 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷 ∈ 𝐶) | |
13 | 11, 12 | sylibr 223 | . . 3 ⊢ (𝜑 → (𝐶 ∩ {𝐷}) = ∅) |
14 | disjiun 4573 | . . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴 ∧ (𝐶 ∩ {𝐷}) = ∅)) → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅) | |
15 | 6, 7, 10, 13, 14 | syl13anc 1320 | . 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ {𝐷}𝐵) = ∅) |
16 | 5, 15 | eqtr3d 2646 | 1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 ∪ 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-ne 2782 df-ral 2901 df-rex 2902 df-rmo 2904 df-v 3175 df-sbc 3403 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-iun 4457 df-disj 4554 |
This theorem is referenced by: caratheodorylem1 39416 |
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