Theorem lttri4

Description: Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Assertion	lttri4	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ltso	⊢ < Or ℝ
2		solin	⊢ ( ( < Or ℝ ∧ ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) → ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) )
3	1 2	mpan	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) )