Theorem ltne

Description: 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999) (Revised by Mario Carneiro, 16-Sep-2015)

	Ref	Expression
Assertion	ltne	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 𝐵 ) → 𝐵 ≠ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ltnr	⊢ ( 𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴 )
2		breq2	⊢ ( 𝐵 = 𝐴 → ( 𝐴 < 𝐵 ↔ 𝐴 < 𝐴 ) )
3	2	notbid	⊢ ( 𝐵 = 𝐴 → ( ¬ 𝐴 < 𝐵 ↔ ¬ 𝐴 < 𝐴 ) )
4	1 3	syl5ibrcom	⊢ ( 𝐴 ∈ ℝ → ( 𝐵 = 𝐴 → ¬ 𝐴 < 𝐵 ) )
5	4	necon2ad	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < 𝐵 → 𝐵 ≠ 𝐴 ) )
6	5	imp	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 𝐵 ) → 𝐵 ≠ 𝐴 )