Theorem nnnlt1

Description: A positive integer is not less than one. (Contributed by NM, 18-Jan-2004) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	nnnlt1	$$⊢A\in \mathbb{N}\to \neg A<1$$

Proof

Step	Hyp	Ref	Expression
1		nnge1	$$⊢A\in \mathbb{N}\to 1\le A$$
2		1re	$$⊢1\in ℝ$$
3		nnre	$$⊢A\in \mathbb{N}\to A\in ℝ$$
4		lenlt	$$⊢\left(1\in ℝ\wedge A\in ℝ\right)\to \left(1\le A\leftrightarrow \neg A<1\right)$$
5	2 3 4	sylancr	$$⊢A\in \mathbb{N}\to \left(1\le A\leftrightarrow \neg A<1\right)$$
6	1 5	mpbid	$$⊢A\in \mathbb{N}\to \neg A<1$$