Theorem ssinss1

Description: Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026)

	Ref	Expression
Assertion	ssinss1	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ssrin	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐴 ∩ 𝐵 ) ⊆ ( 𝐶 ∩ 𝐵 ) )
2		inss1	⊢ ( 𝐶 ∩ 𝐵 ) ⊆ 𝐶
3	1 2	sstrdi	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )