Theorem nelne2

Description: Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012) (Proof shortened by Wolf Lammen, 14-May-2023)

	Ref	Expression
Assertion	nelne2	$$⊢\left(A\in C\wedge \neg B\in C\right)\to A\ne B$$

Proof

Step	Hyp	Ref	Expression
1		nelneq	$$⊢\left(A\in C\wedge \neg B\in C\right)\to \neg A=B$$
2	1	neqned	$$⊢\left(A\in C\wedge \neg B\in C\right)\to A\ne B$$