Theorem xrnltled

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

	Ref	Expression
Hypotheses	xrnltled.1	|- ( ph -> A e. RR* )
	xrnltled.2	|- ( ph -> B e. RR* )
	xrnltled.3	|- ( ph -> -. B < A )
Assertion	xrnltled	|- ( ph -> A <_ B )

Proof

Step	Hyp	Ref	Expression
1		xrnltled.1	|- ( ph -> A e. RR* )
2		xrnltled.2	|- ( ph -> B e. RR* )
3		xrnltled.3	|- ( ph -> -. B < A )
4	1 2	xrlenltd	|- ( ph -> ( A <_ B <-> -. B < A ) )
5	3 4	mpbird	|- ( ph -> A <_ B )