Theorem leloei

Description: 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 14-May-1999)

	Ref	Expression
Hypotheses	lt.1	$$⊢A\in ℝ$$
	lt.2	$$⊢B\in ℝ$$
Assertion	leloei	$$⊢A\le B\leftrightarrow \left(A<B\vee A=B\right)$$

Proof

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