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Mathbox for Thierry Arnoux |
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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 3952 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴) |
4 | 3 | snssd 4804 | . . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝐴) |
5 | 1, 4 | unssd 4178 | . . . 4 ⊢ (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴) |
6 | disjiunel.1 | . . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) | |
7 | disjss1 5109 | . . . 4 ⊢ ((𝐸 ∪ {𝑌}) ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵)) | |
8 | 5, 6, 7 | sylc 65 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵) |
9 | 2 | eldifbd 3953 | . . . 4 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐸) |
10 | disjiunel.2 | . . . . 5 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) | |
11 | 10 | disjunsn 32260 | . . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅))) |
12 | 3, 9, 11 | syl2anc 583 | . . 3 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅))) |
13 | 8, 12 | mpbid 231 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)) |
14 | 13 | simprd 495 | 1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∖ cdif 3937 ∪ cun 3938 ∩ cin 3939 ⊆ wss 3940 ∅c0 4314 {csn 4620 ∪ ciun 4987 Disj wdisj 5103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rmo 3368 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-sn 4621 df-iun 4989 df-disj 5104 |
This theorem is referenced by: disjuniel 32263 |
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