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Metamath Proof Explorer

Theorem letrii

Description: Trichotomy law for 'less than or equal to'. (Contributed by NM, 2-Aug-1999)

Ref Expression
Hypotheses lt.1 ⊢ A ∈ ℝ
lt.2 ⊢ B ∈ ℝ
Assertion letrii ⊢ A ≤ B ∨ B ≤ A

Proof

Step Hyp Ref Expression
1 lt.1 ⊢ A ∈ ℝ
2 lt.2 ⊢ B ∈ ℝ
3 2 1 ltnlei ⊢ B < A ↔ ¬ A ≤ B
4 2 1 ltlei ⊢ B < A → B ≤ A
5 3 4 sylbir ⊢ ¬ A ≤ B → B ≤ A
6 5 orri ⊢ A ≤ B ∨ B ≤ A