TITLE: Cardinality of power set of empty set
QUESTION [8 upvotes]: I'm trying to show that class $C$ of all even-cardinality sets is not closed over powerset via counter-example.
Is it not closed because $|\wp(\{\emptyset\})|=1$ therefore it is not in $C$?
I was wondering mainly if the statement "$|\wp(\{\emptyset\})|=1$ because $\wp\{\emptyset\}=\{\{\emptyset\}\}$" was true.

REPLY [4 votes]: As noted in the comments, $P(\emptyset)=\{{\emptyset\}}$. 
$P(\{{\emptyset\}})=\{{\emptyset,\{{\emptyset\}}\}}\ne\{{\{{\emptyset\}}\}}$