Theorem lttri2

Description: Consequence of trichotomy. (Contributed by NM, 9-Oct-1999)

	Ref	Expression
Assertion	lttri2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≠ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐵 < 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltso	⊢ < Or ℝ
2		sotrieq	⊢ ( ( < 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 ) ) )
4	3	bicomd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ¬ ( 𝐴 < 𝐵 ∨ 𝐵 < 𝐴 ) ↔ 𝐴 = 𝐵 ) )
5	4	necon1abid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≠ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐵 < 𝐴 ) ) )