Is there an operator for 'contained in set or subset or subset of subset' and so on?

Question

Assume I have a set, the elements of which may be sets, some of which may have sets as elements in turn, and so on. Is there an operator or a convenient way to express this relationship?

Answer 1

If I understand you, you're looking for a "recursive membership" operator $\preceq$ which is defined by:

1. $x\in A$ implies that $x \preceq A$.
2. $x \preceq A$ and $A \preceq B$ implies that $x \preceq B$.

In other words, it is the transitive closure of the set membership relation.

I am not sure if a notation for this operator already exists, but this gives you a formal definition to invent it. You might use $\in^*$ for notation, by analogy with Kleene star notation.