Theorem ssini

Description: An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003)

	Ref	Expression
Hypotheses	ssini.1	⊢ 𝐴 ⊆ 𝐵
	ssini.2	⊢ 𝐴 ⊆ 𝐶
Assertion	ssini	⊢ 𝐴 ⊆ ( 𝐵 ∩ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ssini.1	⊢ 𝐴 ⊆ 𝐵
2		ssini.2	⊢ 𝐴 ⊆ 𝐶
3	1 2	pm3.2i	⊢ ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶 )
4		ssin	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶 ) ↔ 𝐴 ⊆ ( 𝐵 ∩ 𝐶 ) )
5	3 4	mpbi	⊢ 𝐴 ⊆ ( 𝐵 ∩ 𝐶 )