Theorem eluz4eluz2

Description: An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023)

	Ref	Expression
Assertion	eluz4eluz2	$$⊢X\in {\mathbb{Z}}_{\ge 4}\to X\in {\mathbb{Z}}_{\ge 2}$$

Proof

Step	Hyp	Ref	Expression
1		2z	$$⊢2\in \mathbb{Z}$$
2		2re	$$⊢2\in ℝ$$
3		4re	$$⊢4\in ℝ$$
4		2lt4	$$⊢2<4$$
5	2 3 4	ltleii	$$⊢2\le 4$$
6		eluzuzle	$$⊢\left(2\in \mathbb{Z}\wedge 2\le 4\right)\to \left(X\in {\mathbb{Z}}_{\ge 4}\to X\in {\mathbb{Z}}_{\ge 2}\right)$$
7	1 5 6	mp2an	$$⊢X\in {\mathbb{Z}}_{\ge 4}\to X\in {\mathbb{Z}}_{\ge 2}$$