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 ) ) |
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 ) ) |