Theorem nelneq

Description: A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997)

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

Proof

Step	Hyp	Ref	Expression
1		eleq1	$$⊢A=B\to \left(A\in C\leftrightarrow B\in C\right)$$
2	1	biimpcd	$$⊢A\in C\to \left(A=B\to B\in C\right)$$
3	2	con3dimp	$$⊢\left(A\in C\wedge \neg B\in C\right)\to \neg A=B$$