Theorem eluzge3nn

Description: If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018)

	Ref	Expression
Assertion	eluzge3nn	$$⊢N\in {\mathbb{Z}}_{\ge 3}\to N\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		1z	$$⊢1\in \mathbb{Z}$$
2		1le3	$$⊢1\le 3$$
3		eluzuzle	$$⊢\left(1\in \mathbb{Z}\wedge 1\le 3\right)\to \left(N\in {\mathbb{Z}}_{\ge 3}\to N\in {\mathbb{Z}}_{\ge 1}\right)$$
4	1 2 3	mp2an	$$⊢N\in {\mathbb{Z}}_{\ge 3}\to N\in {\mathbb{Z}}_{\ge 1}$$
5		elnnuz	$$⊢N\in \mathbb{N}\leftrightarrow N\in {\mathbb{Z}}_{\ge 1}$$
6	4 5	sylibr	$$⊢N\in {\mathbb{Z}}_{\ge 3}\to N\in \mathbb{N}$$