Theorem ltleni

Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999)

	Ref	Expression
Hypotheses	lt.1	$$⊢A\in ℝ$$
	lt.2	$$⊢B\in ℝ$$
Assertion	ltleni	$$⊢A<B\leftrightarrow \left(A\le B\wedge B\ne A\right)$$

Proof

Step	Hyp	Ref	Expression
1		lt.1	$$⊢A\in ℝ$$
2		lt.2	$$⊢B\in ℝ$$
3		ltlen	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A<B\leftrightarrow \left(A\le B\wedge B\ne A\right)\right)$$
4	1 2 3	mp2an	$$⊢A<B\leftrightarrow \left(A\le B\wedge B\ne A\right)$$