Theorem nel1nelini

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

	Ref	Expression
Hypothesis	nel1nelini.1	⊢ ¬ 𝐴 ∈ 𝐵
Assertion	nel1nelini	⊢ ¬ 𝐴 ∈ ( 𝐵 ∩ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		nel1nelini.1	⊢ ¬ 𝐴 ∈ 𝐵
2		nel1nelin	⊢ ( ¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ ( 𝐵 ∩ 𝐶 ) )
3	1 2	ax-mp	⊢ ¬ 𝐴 ∈ ( 𝐵 ∩ 𝐶 )