Theorem eldifsni

Description: Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015)

	Ref	Expression
Assertion	eldifsni	$$⊢A\in \left(B\setminus \left\{C\right\}\right)\to A\ne C$$

Proof

Step	Hyp	Ref	Expression
1		eldifsn	$$⊢A\in \left(B\setminus \left\{C\right\}\right)\leftrightarrow \left(A\in B\wedge A\ne C\right)$$
2	1	simprbi	$$⊢A\in \left(B\setminus \left\{C\right\}\right)\to A\ne C$$