Theorem xrlttri3

Description: Trichotomy law for 'less than' for extended reals. (Contributed by NM, 9-Feb-2006)

	Ref	Expression
Assertion	xrlttri3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		xrltso	⊢ < 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 ) ) )