Theorem ltled

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	ltled	|- ( ph -> A <_ B )

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		ltle	|- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A <_ B ) )
5	1 2 4	syl2anc	|- ( ph -> ( A < B -> A <_ B ) )
6	3 5	mpd	|- ( ph -> A <_ B )