Theorem xrlttri2

Description: Trichotomy law for 'less than' for extended reals. (Contributed by NM, 10-Dec-2007)

	Ref	Expression
Assertion	xrlttri2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≠ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐵 < 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xrltso	⊢ < 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 ) ) )