Theorem lttrd

Description: Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006)

	Ref	Expression
Hypotheses	ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	letrd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	lttrd.4	⊢ ( 𝜑 → 𝐴 < 𝐵 )
	lttrd.5	⊢ ( 𝜑 → 𝐵 < 𝐶 )
Assertion	lttrd	⊢ ( 𝜑 → 𝐴 < 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		letrd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		lttrd.4	⊢ ( 𝜑 → 𝐴 < 𝐵 )
5		lttrd.5	⊢ ( 𝜑 → 𝐵 < 𝐶 )
6		lttr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
7	1 2 3 6	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
8	4 5 7	mp2and	⊢ ( 𝜑 → 𝐴 < 𝐶 )