Theorem ltletri

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

	Ref	Expression
Hypotheses	lt.1	$$⊢A\in ℝ$$
	lt.2	$$⊢B\in ℝ$$
	lt.3	$$⊢C\in ℝ$$
Assertion	ltletri	$$⊢\left(A<B\wedge B\le C\right)\to A<C$$

Proof

Step	Hyp	Ref	Expression
1		lt.1	$$⊢A\in ℝ$$
2		lt.2	$$⊢B\in ℝ$$
3		lt.3	$$⊢C\in ℝ$$
4		ltletr	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(\left(A<B\wedge B\le C\right)\to A<C\right)$$
5	1 2 3 4	mp3an	$$⊢\left(A<B\wedge B\le C\right)\to A<C$$