Theorem inelros

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

	Ref	Expression
Hypothesis	isros.1	⊢ 𝑄 = { 𝑠 ∈ 𝒫 𝒫 𝑂 ∣ ( ∅ ∈ 𝑠 ∧ ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑠 ( ( 𝑥 ∪ 𝑦 ) ∈ 𝑠 ∧ ( 𝑥 ∖ 𝑦 ) ∈ 𝑠 ) ) }
Assertion	inelros	⊢ ( ( 𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ∩ 𝐵 ) ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		isros.1	⊢ 𝑄 = { 𝑠 ∈ 𝒫 𝒫 𝑂 ∣ ( ∅ ∈ 𝑠 ∧ ∀ 𝑥 ∈ 𝑠 ∀ 𝑦 ∈ 𝑠 ( ( 𝑥 ∪ 𝑦 ) ∈ 𝑠 ∧ ( 𝑥 ∖ 𝑦 ) ∈ 𝑠 ) ) }
2		dfin4	⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) )
3	1	difelros	⊢ ( ( 𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ∖ 𝐵 ) ∈ 𝑆 )
4	1	difelros	⊢ ( ( 𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ ( 𝐴 ∖ 𝐵 ) ∈ 𝑆 ) → ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) ) ∈ 𝑆 )
5	3 4	syld3an3	⊢ ( ( 𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ∖ ( 𝐴 ∖ 𝐵 ) ) ∈ 𝑆 )
6	2 5	eqeltrid	⊢ ( ( 𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ∩ 𝐵 ) ∈ 𝑆 )