Theorem nel2nelini

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

	Ref	Expression
Hypothesis	nel2nelini.1	⊢ ¬ 𝐴 ∈ 𝐶
Assertion	nel2nelini	⊢ ¬ 𝐴 ∈ ( 𝐵 ∩ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		nel2nelini.1	⊢ ¬ 𝐴 ∈ 𝐶
2		nel2nelin	⊢ ( ¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) )
3	1 2	ax-mp	⊢ ¬ 𝐴 ∈ ( 𝐵 ∩ 𝐶 )