Theorem nnnle0

Description: A positive integer is not less than or equal to zero. (Contributed by AV, 13-May-2020)

	Ref	Expression
Assertion	nnnle0	$$⊢A\in \mathbb{N}\to \neg A\le 0$$

Proof

Step	Hyp	Ref	Expression
1		nngt0	$$⊢A\in \mathbb{N}\to 0<A$$
2		0re	$$⊢0\in ℝ$$
3		nnre	$$⊢A\in \mathbb{N}\to A\in ℝ$$
4		ltnle	$$⊢\left(0\in ℝ\wedge A\in ℝ\right)\to \left(0<A\leftrightarrow \neg A\le 0\right)$$
5	4	bicomd	$$⊢\left(0\in ℝ\wedge A\in ℝ\right)\to \left(\neg A\le 0\leftrightarrow 0<A\right)$$
6	2 3 5	sylancr	$$⊢A\in \mathbb{N}\to \left(\neg A\le 0\leftrightarrow 0<A\right)$$
7	1 6	mpbird	$$⊢A\in \mathbb{N}\to \neg A\le 0$$