Theorem lelttri

Description: 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999)

	Ref	Expression
Hypotheses	lt.1	|- A e. RR
	lt.2	|- B e. RR
	lt.3	|- C e. RR
Assertion	lelttri	|- ( ( A <_ B /\ B < C ) -> A < C )

Proof

Step	Hyp	Ref	Expression
1		lt.1	|- A e. RR
2		lt.2	|- B e. RR
3		lt.3	|- C e. RR
4		lelttr	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ B /\ B < C ) -> A < C ) )
5	1 2 3 4	mp3an	|- ( ( A <_ B /\ B < C ) -> A < C )