Cardinality of power set of empty set

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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.

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1 Answer

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As noted in the comments, $P(\emptyset)=\{{\emptyset\}}$.

$P(\{{\emptyset\}})=\{{\emptyset,\{{\emptyset\}}\}}\ne\{{\{{\emptyset\}}\}}$

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