Theorem xrgtned

Description: 'Greater than' implies not equal. (Contributed by Glauco Siliprandi, 17-Aug-2020)

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

Proof

Step	Hyp	Ref	Expression
1		xrgtned.1	$$⊢\phi \to A\in {ℝ}^{*}$$
2		xrgtned.2	$$⊢\phi \to B\in {ℝ}^{*}$$
3		xrgtned.3	$$⊢\phi \to A<B$$
4		xrltne	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge A<B\right)\to B\ne A$$
5	1 2 3 4	syl3anc	$$⊢\phi \to B\ne A$$