Theorem ltnlei

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

	Ref	Expression
Hypotheses	lt.1	⊢ 𝐴 ∈ ℝ
	lt.2	⊢ 𝐵 ∈ ℝ
Assertion	ltnlei	⊢ ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		lt.1	⊢ 𝐴 ∈ ℝ
2		lt.2	⊢ 𝐵 ∈ ℝ
3	2 1	lenlti	⊢ ( 𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵 )
4	3	con2bii	⊢ ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 )