Theorem elini

Description: Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	elini.1	⊢ 𝐴 ∈ 𝐵
	elini.2	⊢ 𝐴 ∈ 𝐶
Assertion	elini	⊢ 𝐴 ∈ ( 𝐵 ∩ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		elini.1	⊢ 𝐴 ∈ 𝐵
2		elini.2	⊢ 𝐴 ∈ 𝐶
3		elin	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) )
4	1 2 3	mpbir2an	⊢ 𝐴 ∈ ( 𝐵 ∩ 𝐶 )