If I wanted to write, there exists an integer $n$ then in symbols this would be written as $$\exists n\in\mathbb{Z}.$$ If I wanted to write, there exists only one integer $n$ then in symbols would be written as $$\exists!n\in\mathbb{Z}.$$ Sometimes this could be mistaken as there exists $!n$ but if so that would be written as $$\exists(!n)\ \lor \ \exists\{!n\}.$$ But what if I wanted to write there exists infinitely many integers $n$. I can’t write $$\exists \text{ infinite } n \in\mathbb{Z}$$ or perhaps $$\exists\text{ inf }n\in\mathbb{Z}$$ can I? If not, is there some mathematical notation to denote this? I thought of something like $$\operatorname*{\text{Exi}}\limits_{n\to\infty}\tag{inspired by $\lim_{n\to\infty}n$}n\in\mathbb{Z}$$ but before I make something up, I want to know for sure if there does not exist a notation of this sort yet. This question came to me after looking at this post.
Thank you in advance.
You mean something like : $$ \exists A \in \mathcal{P}(\mathbb{Z}), \ |A|=\infty, \quad \forall n \in A, \ \cdots $$ (or card$(A)=\infty$)?