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Metamath Proof Explorer

Theorem xrnltled

Description: "Not less than" implies "less than or equal to". (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses xrnltled.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
xrnltled.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
xrnltled.3 ⊢ ( 𝜑 → ¬ 𝐵 < 𝐴 )
Assertion xrnltled ( 𝜑 → 𝐴 ≤ 𝐵 )

Proof

Step Hyp Ref Expression
1 xrnltled.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2 xrnltled.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3 xrnltled.3 ⊢ ( 𝜑 → ¬ 𝐵 < 𝐴 )
4 1 2 xrlenltd ⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
5 3 4 mpbird ⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )