For the purposes of something I'm working on, I need to define and use notation referring to the elements of sets which are themselves elements of another set, to an arbitrary depth, and that only appear once.
I've seen the notation $x \in_n X$ used with multisets, to denote that the element $x$ has a multiplicity of $n$ in the multiset $X$, so I figure I could use $\in_1^k$ and then define it alongside the formula. It's just cumbersome to use language like "an element of an element of an element of..." to a depth of $k$. Even "$x$ appears exactly once in the nested elements of $X$ at a depth of $k$" may be hard to understand. Maybe, "$x$ is a $k$th-order sub-element of $X$ with a multiplicity of $1$"?
Is there a more concise way I can describe this? Or, even better, are there already notations or terms that would fit this purpose? Although I kind of like my last idea. Is that an intuitive phrasing?