Theorem ltnsymd

Description: 'Less than' implies '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 )
	ltled.1	|- ( ph -> A < B )
Assertion	ltnsymd	|- ( ph -> -. B < A )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	|- ( ph -> A e. RR )
2		ltd.2	|- ( ph -> B e. RR )
3		ltled.1	|- ( ph -> A < B )
4	1 2 3	ltled	|- ( ph -> A <_ B )
5	1 2	lenltd	|- ( ph -> ( A <_ B <-> -. B < A ) )
6	4 5	mpbid	|- ( ph -> -. B < A )