Theorem ltnle

Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005)

	Ref	Expression
Assertion	ltnle	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		lenlt	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵 ) )
2	1	ancoms	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵 ) )
3	2	con2bid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) )