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Metamath Proof Explorer

Theorem nltled

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

Ref Expression
Hypotheses ltd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
nltled.1 ⊢ ( 𝜑 → ¬ 𝐵 < 𝐴 )
Assertion nltled ( 𝜑 → 𝐴 ≤ 𝐵 )

Proof

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