Theorem ssrind

Description: Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypothesis	ssrind.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
Assertion	ssrind	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) ⊆ ( 𝐵 ∩ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssrind.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		ssrin	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐶 ) ⊆ ( 𝐵 ∩ 𝐶 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) ⊆ ( 𝐵 ∩ 𝐶 ) )