Theorem gtneii

Description: 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013)

	Ref	Expression
Hypotheses	lt.1	$$⊢A\in ℝ$$
	ltneii.2	$$⊢A<B$$
Assertion	gtneii	$$⊢B\ne A$$

Proof

Step	Hyp	Ref	Expression
1		lt.1	$$⊢A\in ℝ$$
2		ltneii.2	$$⊢A<B$$
3		ltne	$$⊢\left(A\in ℝ\wedge A<B\right)\to B\ne A$$
4	1 2 3	mp2an	$$⊢B\ne A$$