Theorem gtneii

Description: 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013)

	Ref	Expression
Hypotheses	lt.1	⊢ 𝐴 ∈ ℝ
	ltneii.2	⊢ 𝐴 < 𝐵
Assertion	gtneii	⊢ 𝐵 ≠ 𝐴

Proof

Step	Hyp	Ref	Expression
1		lt.1	⊢ 𝐴 ∈ ℝ
2		ltneii.2	⊢ 𝐴 < 𝐵
3		ltne	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 𝐵 ) → 𝐵 ≠ 𝐴 )
4	1 2 3	mp2an	⊢ 𝐵 ≠ 𝐴