Notation for elements nested within elements of a set?

Question

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?

Answer 1

Let $\cup^0 X=X$ and $\cup^{n+1}X=\cup (\cup^nX).$ The transitive closure of X is tr cl $X=\cup_{0\le n\in \Bbb Z}(\cup^n X)....$ (A set $Y$ is transitive iff $\forall y\in Y \,(y\subset Y).$ And tr cl $X$ is the $\subset$-smallest transitive $Y$ such that $X\subset Y.)$

For $x\in$ tr cl $X$ let $d_X(x)$ be the least $n$ such that $x\in \cup^n X.$ ($d$ for depth.) And let $\text { Lev}_X(n)=\{x: d_X(x)=n\}.$ (Lev for level.)

Note that if $d_X(x)=n>0$ then $x$ may belong to two or more members of Lev$_X(n-1).$

I have seen the notation $\cup^n$ in a textbook $[1]$, and tr cl $X$ is standard, although with stylistic variety, e.g. TrCl($X$), but to be safe it is probably best to give their definitions when you are writing.

$[1].$ Kunen, Kenneth. Set Theory : An Introduction To Independence Proofs.

Answer 2

If K is a collection of sets, then x in $\cup$K states that x is an element of one of the sets in K.

x in A in K states that x is an element of the set A of the collection K.