This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) )

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 ) ) )