Theorem xrnltled

Description: "Not less than" implies "less than or equal to". (Contributed by Glauco Siliprandi, 11-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		xrnltled.1	$$⊢\phi \to A\in {ℝ}^{*}$$
2		xrnltled.2	$$⊢\phi \to B\in {ℝ}^{*}$$
3		xrnltled.3	$$⊢\phi \to \neg B<A$$
4	1 2	xrlenltd	$$⊢\phi \to \left(A\le B\leftrightarrow \neg B<A\right)$$
5	3 4	mpbird	$$⊢\phi \to A\le B$$