Theorem sslin

Description: Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999)

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

Proof

Step	Hyp	Ref	Expression
1		ssrin	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐶 ) ⊆ ( 𝐵 ∩ 𝐶 ) )
2		incom	⊢ ( 𝐶 ∩ 𝐴 ) = ( 𝐴 ∩ 𝐶 )
3		incom	⊢ ( 𝐶 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐶 )
4	1 2 3	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∩ 𝐴 ) ⊆ ( 𝐶 ∩ 𝐵 ) )