Theorem elinel1

Description: Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	elinel1	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) → 𝐴 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elin	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ) )
2	1	simplbi	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) → 𝐴 ∈ 𝐵 )