Theorem lttri3

Description: Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999)

	Ref	Expression
Assertion	lttri3	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltso	⊢ < Or ℝ
2		sotrieq2	⊢ ( ( < 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 ) ) )