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	$$⊢\neg A\in B$$
Assertion	nel1nelini	$$⊢\neg A\in \left(B\cap C\right)$$

Proof

Step	Hyp	Ref	Expression
1		nel1nelini.1	$$⊢\neg A\in B$$
2		nel1nelin	$$⊢\neg A\in B\to \neg A\in \left(B\cap C\right)$$
3	1 2	ax-mp	$$⊢\neg A\in \left(B\cap C\right)$$