Theorem ltnsymd

Description: 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		ltled.1	⊢ ( 𝜑 → 𝐴 < 𝐵 )
4	1 2 3	ltled	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
5	1 2	lenltd	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
6	4 5	mpbid	⊢ ( 𝜑 → ¬ 𝐵 < 𝐴 )