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	$$⊢\left(A\in ℝ\wedge A<B\right)\to B\ne A$$

Proof

Step	Hyp	Ref	Expression
1		ltnr	$$⊢A\in ℝ\to \neg A<A$$
2		breq2	$$⊢B=A\to \left(A<B\leftrightarrow A<A\right)$$
3	2	notbid	$$⊢B=A\to \left(\neg A<B\leftrightarrow \neg A<A\right)$$
4	1 3	syl5ibrcom	$$⊢A\in ℝ\to \left(B=A\to \neg A<B\right)$$
5	4	necon2ad	$$⊢A\in ℝ\to \left(A<B\to B\ne A\right)$$
6	5	imp	$$⊢\left(A\in ℝ\wedge A<B\right)\to B\ne A$$