Theorem nel1nelin

Description: Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Assertion	nel1nelin	⊢ ( ¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		elinel1	⊢ ( 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) → 𝐴 ∈ 𝐵 )
2	1	con3i	⊢ ( ¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) )