Your usage of for all, and such that, is fine; the inequality is not fine, since it suggests $-3\gt 3$ which is not so. There are a few ways to write what you’re saying in a better way:
$$\forall n\in\{-4,-5,\cdots\}\cup\{4,5,\cdots\},\,|D_n|=0$$
Or:
$$\forall n\in\Bbb Z,\,|n|\gt3,\,|D_n|=0$$
Or:
$$\forall n\in\Bbb Z\setminus\{-3,-2,\cdots,3\},\,|D_n|=0$$
Or:
$$\forall n\in\Bbb Z:n\lt-3\vee n\gt3,\,|D_n|=0$$
Where \vee
$\vee$ is the logical OR.
Notice that it is not even necessary (but of course you can) to write “such that”. Sometimes we abbreviate such that, or imply it grammatically, by using a colon.
The $\ni$ notation can be used for “such that”, but this is not very common in my experience. It is much more common to use it to denote set membership, as $\in$ does, except that the author is trying to emphasise the set which comes first, or perhaps that it makes more sense to let things flow in the $\ni$ direction than the $\in$.
Example:
You might want to say that $U$ is a set containing a point $x_0$, but here you are maybe letting $U$ be an arbitrary set with some conditions plus the condition that $x_0\in U$. However, in the sentence structure it can be more natural to write (e.g.):
“There exists an open $U\ni x_0$”
As opposed to:
“There exists an open $U$, with $x_0\in U$”
Another use case is in function declaration, where the author does not care to give the map a name as such since this is not the important; they rather want to show what this map will do and what properties it has. That is, it’s common to write $f(x)=x^2$ as a function definition, especially lower down the school, but more abstractly an author might write:
“Since the map: $$\Bbb R\ni x\mapsto x^2\in\Bbb R$$Is continuous...”
And the rest of the sentence would maybe make reference to the map, to the function, but it was never necessary to give the function a name (e.g. $f$). This notation also shows where $x$ comes from (what kind of number/object it is) and it can flow better to use $\ni$ there rather than:
“Since the map: $$\Bbb R\to\Bbb R,\,x\mapsto x^2$$Is continuous...”
As this is maybe longer or not preferable for some authors.