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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjxun0 | Structured version Visualization version GIF version |
Description: Simplify a disjoint union. (Contributed by Thierry Arnoux, 27-Nov-2023.) |
Ref | Expression |
---|---|
disjxun0.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = ∅) |
Ref | Expression |
---|---|
disjxun0 | ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjxun0.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = ∅) | |
2 | nel02 4331 | . . . . 5 ⊢ (𝐶 = ∅ → ¬ 𝑦 ∈ 𝐶) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑦 ∈ 𝐶) |
4 | 3 | rmounid 32002 | . . 3 ⊢ (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
5 | 4 | albidv 1921 | . 2 ⊢ (𝜑 → (∀𝑦∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
6 | df-disj 5113 | . 2 ⊢ (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∀𝑦∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶) | |
7 | df-disj 5113 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
8 | 5, 6, 7 | 3bitr4g 313 | 1 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1537 = wceq 1539 ∈ wcel 2104 ∃*wrmo 3373 ∪ cun 3945 ∅c0 4321 Disj wdisj 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-mo 2532 df-clab 2708 df-cleq 2722 df-clel 2808 df-rmo 3374 df-v 3474 df-dif 3950 df-un 3952 df-nul 4322 df-disj 5113 |
This theorem is referenced by: tocyccntz 32573 |
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