Theorem lelttrd

Description: Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006)

	Ref	Expression
Hypotheses	ltd.1	$$⊢\phi \to A\in ℝ$$
	ltd.2	$$⊢\phi \to B\in ℝ$$
	letrd.3	$$⊢\phi \to C\in ℝ$$
	lelttrd.4	$$⊢\phi \to A\le B$$
	lelttrd.5	$$⊢\phi \to B<C$$
Assertion	lelttrd	$$⊢\phi \to A<C$$

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ltd.2	$$⊢\phi \to B\in ℝ$$
3		letrd.3	$$⊢\phi \to C\in ℝ$$
4		lelttrd.4	$$⊢\phi \to A\le B$$
5		lelttrd.5	$$⊢\phi \to B<C$$
6		lelttr	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(\left(A\le B\wedge B<C\right)\to A<C\right)$$
7	1 2 3 6	syl3anc	$$⊢\phi \to \left(\left(A\le B\wedge B<C\right)\to A<C\right)$$
8	4 5 7	mp2and	$$⊢\phi \to A<C$$