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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjiunel | 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 |
---|---|
disjiunel.1 | ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) |
disjiunel.2 | ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) |
disjiunel.3 | ⊢ (𝜑 → 𝐸 ⊆ 𝐴) |
disjiunel.4 | ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 𝐸)) |
Ref | Expression |
---|---|
disjiunel | ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjiunel.3 | . . . . 5 ⊢ (𝜑 → 𝐸 ⊆ 𝐴) | |
2 | disjiunel.4 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 𝐸)) | |
3 | 2 | eldifad 3552 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴) |
4 | 3 | snssd 4281 | . . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝐴) |
5 | 1, 4 | unssd 3751 | . . . 4 ⊢ (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴) |
6 | disjiunel.1 | . . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) | |
7 | disjss1 4559 | . . . . 5 ⊢ ((𝐸 ∪ {𝑌}) ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵)) | |
8 | 7 | imp 444 | . . . 4 ⊢ (((𝐸 ∪ {𝑌}) ⊆ 𝐴 ∧ Disj 𝑥 ∈ 𝐴 𝐵) → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵) |
9 | 5, 6, 8 | syl2anc 691 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵) |
10 | 2 | eldifbd 3553 | . . . 4 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐸) |
11 | disjiunel.2 | . . . . 5 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) | |
12 | 11 | disjunsn 28789 | . . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅))) |
13 | 3, 10, 12 | syl2anc 691 | . . 3 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅))) |
14 | 9, 13 | mpbid 221 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)) |
15 | 14 | simprd 478 | 1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 ∩ 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-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-iun 4457 df-disj 4554 |
This theorem is referenced by: disjuniel 28792 |
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