Question: is a set an element of its power set

Answer: Yes, In mathematics, the power set (or powerset) of any set S is the set of all subsets of S, including the empty set and S itself, variously denoted as P(S), P(S), ℘(S) (using the ``Weierstrass p''), P(S), P(S), or, identifying the powerset of S with the set of all functions from S to a given set of two elements, 2. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.