Theorem gtned

Description: 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	$$⊢\phi \to A\in ℝ$$
	ltned.2	$$⊢\phi \to A<B$$
Assertion	gtned	$$⊢\phi \to B\ne A$$

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ltned.2	$$⊢\phi \to A<B$$
3		ltne	$$⊢\left(A\in ℝ\wedge A<B\right)\to B\ne A$$
4	1 2 3	syl2anc	$$⊢\phi \to B\ne A$$