Theorem xrltned

Description: 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020)

	Ref	Expression
Hypotheses	xrltned.1	$$⊢\phi \to A\in {ℝ}^{*}$$
	xrltned.2	$$⊢\phi \to B\in {ℝ}^{*}$$
	xrltned.3	$$⊢\phi \to A<B$$
Assertion	xrltned	$$⊢\phi \to A\ne B$$

Proof

Step	Hyp	Ref	Expression
1		xrltned.1	$$⊢\phi \to A\in {ℝ}^{*}$$
2		xrltned.2	$$⊢\phi \to B\in {ℝ}^{*}$$
3		xrltned.3	$$⊢\phi \to A<B$$
4	1 2 3	xrgtned	$$⊢\phi \to B\ne A$$
5	4	necomd	$$⊢\phi \to A\ne B$$