Theorem nn0nlt0

Description: A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	nn0nlt0	$$⊢A\in {\mathbb{N}}_0\to \neg A<0$$

Proof

Step	Hyp	Ref	Expression
1		nn0ge0	$$⊢A\in {\mathbb{N}}_0\to 0\le A$$
2		0re	$$⊢0\in ℝ$$
3		nn0re	$$⊢A\in {\mathbb{N}}_0\to A\in ℝ$$
4		lenlt	$$⊢\left(0\in ℝ\wedge A\in ℝ\right)\to \left(0\le A\leftrightarrow \neg A<0\right)$$
5	2 3 4	sylancr	$$⊢A\in {\mathbb{N}}_0\to \left(0\le A\leftrightarrow \neg A<0\right)$$
6	1 5	mpbid	$$⊢A\in {\mathbb{N}}_0\to \neg A<0$$