Theorem ssinss1d

Description: Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypothesis	ssinss1d.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )
Assertion	ssinss1d	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ssinss1d.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )
2		ssinss1	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )