Theorem nelne2OLD

Description: Obsolete version of nelne2 asw of 14-May-2023. (Contributed by NM, 25-Jun-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	nelne2OLD	$$⊢\left(A\in C\wedge \neg B\in C\right)\to A\ne B$$

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	necon3bd	$$⊢A\in C\to \left(\neg B\in C\to A\ne B\right)$$
4	3	imp	$$⊢\left(A\in C\wedge \neg B\in C\right)\to A\ne B$$