Theorem lensymd

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

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

Proof

Step	Hyp	Ref	Expression
1		ltd.1	|- ( ph -> A e. RR )
2		ltd.2	|- ( ph -> B e. RR )
3		lensymd.3	|- ( ph -> A <_ B )
4	1 2	lenltd	|- ( ph -> ( A <_ B <-> -. B < A ) )
5	3 4	mpbid	|- ( ph -> -. B < A )