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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 )