Theorem lensymd

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

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

Proof

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