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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