Theorem uzct

Description: An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypothesis	uzct.1	$$⊢Z={\mathbb{Z}}_{\ge N}$$
Assertion	uzct	$$⊢Z\preccurlyeq \mathrm{\omega }$$

Proof

Step	Hyp	Ref	Expression
1		uzct.1	$$⊢Z={\mathbb{Z}}_{\ge N}$$
2		uzssz	$$⊢{\mathbb{Z}}_{\ge N}\subseteq \mathbb{Z}$$
3	1 2	eqsstri	$$⊢Z\subseteq \mathbb{Z}$$
4		zex	$$⊢\mathbb{Z}\in \mathrm{V}$$
5		ssdomg	$$⊢\mathbb{Z}\in \mathrm{V}\to \left(Z\subseteq \mathbb{Z}\to Z\preccurlyeq \mathbb{Z}\right)$$
6	4 5	ax-mp	$$⊢Z\subseteq \mathbb{Z}\to Z\preccurlyeq \mathbb{Z}$$
7	3 6	ax-mp	$$⊢Z\preccurlyeq \mathbb{Z}$$
8		zct	$$⊢\mathbb{Z}\preccurlyeq \mathrm{\omega }$$
9		domtr	$$⊢\left(Z\preccurlyeq \mathbb{Z}\wedge \mathbb{Z}\preccurlyeq \mathrm{\omega }\right)\to Z\preccurlyeq \mathrm{\omega }$$
10	7 8 9	mp2an	$$⊢Z\preccurlyeq \mathrm{\omega }$$