Metamath Proof Explorer

Theorem xrlelttric

Description: Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017)

Ref Expression
Assertion xrlelttric ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴 ) )

Proof

Step Hyp Ref Expression
1 pm2.1 ⊢ ( ¬ 𝐵 < 𝐴 ∨ 𝐵 < 𝐴 )
2 xrlenlt ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
3 2 orbi1d ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴 ) ↔ ( ¬ 𝐵 < 𝐴 ∨ 𝐵 < 𝐴 ) ) )
4 1 3 mpbiri ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴 ) )