Theorem eluz4eluz3

Description: An integer greater than or equal to 4 is an integer greater than or equal to 3. (Contributed by AV, 5-Sep-2025)

	Ref	Expression
Assertion	eluz4eluz3	$$⊢X\in {\mathbb{Z}}_{\ge 4}\to X\in {\mathbb{Z}}_{\ge 3}$$

Proof

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