Theorem ltnei

Description: 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999)

	Ref	Expression
Hypotheses	lt.1	⊢ 𝐴 ∈ ℝ
	lt.2	⊢ 𝐵 ∈ ℝ
Assertion	ltnei	⊢ ( 𝐴 < 𝐵 → 𝐵 ≠ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		lt.1	⊢ 𝐴 ∈ ℝ
2		lt.2	⊢ 𝐵 ∈ ℝ
3		ltne	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 𝐵 ) → 𝐵 ≠ 𝐴 )
4	1 3	mpan	⊢ ( 𝐴 < 𝐵 → 𝐵 ≠ 𝐴 )