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 )