Theorem xrltnled

Description: 'Less than' in terms of 'less than or equal to'. (Contributed by Glauco Siliprandi, 3-Mar-2021)

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

Proof

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