Theorem inss

Description: Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014)

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

Proof

Step	Hyp	Ref	Expression
1		ssinss1	⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )
2		incom	⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 )
3		ssinss1	⊢ ( 𝐵 ⊆ 𝐶 → ( 𝐵 ∩ 𝐴 ) ⊆ 𝐶 )
4	2 3	eqsstrid	⊢ ( 𝐵 ⊆ 𝐶 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )
5	1 4	jaoi	⊢ ( ( 𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶 ) → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐶 )