Theorem inelros 34917

Description: A ring of sets is closed under intersection. (Contributed by Thierry Arnoux, 19-Jul-2020.)

Hypothesis

Ref	Expression
isros.1	⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}

Assertion

Ref	Expression
inelros	⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)

Distinct variable groups:   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑠)   𝐵(𝑥, 𝑦, 𝑠)   𝑄(𝑥, 𝑦, 𝑠)   𝑂(𝑥, 𝑦)

Proof of Theorem inelros

Step	Hyp	Ref	Expression
1		dfin4 4226	. 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 34916	. . 3 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)
4	2	difelros 34916	. . 3 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → (𝐴 ∖ (𝐴 ∖ 𝐵)) ∈ 𝑆)
5	3, 4	syld3an3 1437	. 2 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ (𝐴 ∖ 𝐵)) ∈ 𝑆)
6	1, 5	eqeltrid 2868	1 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  𝒫 cpw 4560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154

This theorem is used by:  rossros  34924