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Metamath Proof Explorer

Theorem lelttric

Description: Trichotomy law. (Contributed by NM, 4-Apr-2005)

Ref Expression
Assertion lelttric ⊢ A ∈ ℝ ∧ B ∈ ℝ → A ≤ B ∨ B < A

Proof

Step Hyp Ref Expression
1 pm2.1 ⊢ ¬ B < A ∨ B < A
2 lenlt ⊢ 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