Theorem neldif

Description: Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994)

	Ref	Expression
Assertion	neldif	$$⊢\left(A\in B\wedge \neg A\in \left(B\setminus C\right)\right)\to A\in C$$

Proof

Step	Hyp	Ref	Expression
1		eldif	$$⊢A\in \left(B\setminus C\right)\leftrightarrow \left(A\in B\wedge \neg A\in C\right)$$
2	1	simplbi2	$$⊢A\in B\to \left(\neg A\in C\to A\in \left(B\setminus C\right)\right)$$
3	2	con1d	$$⊢A\in B\to \left(\neg A\in \left(B\setminus C\right)\to A\in C\right)$$
4	3	imp	$$⊢\left(A\in B\wedge \neg A\in \left(B\setminus C\right)\right)\to A\in C$$