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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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    The empty set is $\varnothing$; it has no elements. $\{\varnothing\}$ is the set whose only member is the empty set, so it has one element. Where you write $\wp\{\varnothing\}$, you mean $\wp(\varnothing)$, the set of all subsets of the empty set; this is $\{\varnothing\}$, the one-element set whose only member is the empty set.2012-09-10

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

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