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 )