Theorem lttri

Description: 'Less than' is transitive. Theorem I.17 of Apostol p. 20. (Contributed by NM, 14-May-1999)

	Ref	Expression
Hypotheses	lt.1	⊢ 𝐴 ∈ ℝ
	lt.2	⊢ 𝐵 ∈ ℝ
	lt.3	⊢ 𝐶 ∈ ℝ
Assertion	lttri	⊢ ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		lt.1	⊢ 𝐴 ∈ ℝ
2		lt.2	⊢ 𝐵 ∈ ℝ
3		lt.3	⊢ 𝐶 ∈ ℝ
4		lttr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
5	1 2 3 4	mp3an	⊢ ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 )