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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 3586 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴) |
4 | 3 | snssd 4340 | . . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝐴) |
5 | 1, 4 | unssd 3789 | . . . 4 ⊢ (𝜑 → (𝐸 ∪ {𝑌}) ⊆ 𝐴) |
6 | disjiunel.1 | . . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) | |
7 | disjss1 4626 | . . . 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 3587 | . . . 4 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐸) |
10 | disjiunel.2 | . . . . 5 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) | |
11 | 10 | disjunsn 29407 | . . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐸) → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅))) |
12 | 3, 9, 11 | syl2anc 693 | . . 3 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐸 ∪ {𝑌})𝐵 ↔ (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅))) |
13 | 8, 12 | mpbid 222 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐸 𝐵 ∧ (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅)) |
14 | 13 | simprd 479 | 1 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐸 𝐵 ∩ 𝐷) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 ∪ ciun 4520 Disj wdisj 4620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-iun 4522 df-disj 4621 |
This theorem is referenced by: disjuniel 29410 |
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