Theorem lenlt

Description: 'Less than or equal to' expressed in terms of 'less than'. (Contributed by NM, 13-May-1999)

	Ref	Expression
Assertion	lenlt	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\le B\leftrightarrow \neg B<A\right)$$

Proof

Step	Hyp	Ref	Expression
1		rexr	$$⊢A\in ℝ\to A\in {ℝ}^{*}$$
2		rexr	$$⊢B\in ℝ\to B\in {ℝ}^{*}$$
3		xrlenlt	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A\le B\leftrightarrow \neg B<A\right)$$
4	1 2 3	syl2an	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\le B\leftrightarrow \neg B<A\right)$$