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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 3960 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴) |
4 | 3 | snssd 4812 | . . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝐴) |
5 | 1, 4 | unssd 4186 | . . . 4 ⊢ (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴) |
6 | disjiunel.1 | . . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) | |
7 | disjss1 5119 | . . . 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 3961 | . . . 4 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐸) |
10 | disjiunel.2 | . . . . 5 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) | |
11 | 10 | disjunsn 32093 | . . . 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 1540 ∈ wcel 2105 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 {csn 4628 ∪ ciun 4997 Disj wdisj 5113 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rmo 3375 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-sn 4629 df-iun 4999 df-disj 5114 |
This theorem is referenced by: disjuniel 32096 |
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