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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > inelros | Structured version Visualization version GIF version |
Description: A ring of sets is closed under intersection. (Contributed by Thierry Arnoux, 19-Jul-2020.) |
Ref | Expression |
---|---|
isros.1 | ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))} |
Ref | Expression |
---|---|
inelros | ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin4 4231 | . 2 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) | |
2 | isros.1 | . . . 4 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))} | |
3 | 2 | difelros 32835 | . . 3 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆) |
4 | 2 | difelros 32835 | . . 3 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → (𝐴 ∖ (𝐴 ∖ 𝐵)) ∈ 𝑆) |
5 | 3, 4 | syld3an3 1410 | . 2 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ (𝐴 ∖ 𝐵)) ∈ 𝑆) |
6 | 1, 5 | eqeltrid 2838 | 1 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 {crab 3406 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ∅c0 4286 𝒫 cpw 4564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 |
This theorem is referenced by: rossros 32843 |
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