Theorem ltnled

Description: 'Less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016)

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

Proof

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