Theorem ssinss1

Description: Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999)

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

Proof

Step	Hyp	Ref	Expression
1		inss1	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴
2		sstr2	⊢ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 → ( 𝐴 ⊆ 𝐶 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )