Theorem xrlenltd

Description: "Less than or equal to" expressed in terms of "less than", for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	xrlenltd.a	|- ( ph -> A e. RR* )
	xrlenltd.b	|- ( ph -> B e. RR* )
Assertion	xrlenltd	|- ( ph -> ( A <_ B <-> -. B < A ) )

Proof

Step	Hyp	Ref	Expression
1		xrlenltd.a	|- ( ph -> A e. RR* )
2		xrlenltd.b	|- ( ph -> B e. RR* )
3		xrlenlt	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
4	1 2 3	syl2anc	|- ( ph -> ( A <_ B <-> -. B < A ) )