Description: Trichotomy law for extended reals. (Contributed by NM, 7-Feb-2007)
Ref | Expression | ||
---|---|---|---|
Assertion | xrletri | ⊢ ( ( 𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*} ) → ( 𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltnle | ⊢ ( ( 𝐵 ∈ ℝ^{*} ∧ 𝐴 ∈ ℝ^{*} ) → ( 𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵 ) ) | |
2 | 1 | ancoms | ⊢ ( ( 𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*} ) → ( 𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵 ) ) |
3 | xrltle | ⊢ ( ( 𝐵 ∈ ℝ^{*} ∧ 𝐴 ∈ ℝ^{*} ) → ( 𝐵 < 𝐴 → 𝐵 ≤ 𝐴 ) ) | |
4 | 3 | ancoms | ⊢ ( ( 𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*} ) → ( 𝐵 < 𝐴 → 𝐵 ≤ 𝐴 ) ) |
5 | 2 4 | sylbird | ⊢ ( ( 𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*} ) → ( ¬ 𝐴 ≤ 𝐵 → 𝐵 ≤ 𝐴 ) ) |
6 | 5 | orrd | ⊢ ( ( 𝐴 ∈ ℝ^{*} ∧ 𝐵 ∈ ℝ^{*} ) → ( 𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴 ) ) |