I'm wondering wether the class $C$ defined by: $$x\in C\Leftrightarrow x\subseteq C\quad\quad (1)$$ Is unique. Its existence is ensured by the following argument:
Let $U$ be the class of all sets, and $x\in U$ a set in it. Since $U$ contains all sets, then it must contain $\{x\}$, and so: $$x\subseteq\{x\}\in U\Rightarrow x\subseteq U$$
Likewise, if $x\subseteq U$ is a set, it is, by definition, a member of $U$ ($U$ is non-empty).
So the class of all sets fits the definition of (1). My question is, is $U$ the only set that fits this description? By the axiom of extensionality we know $U$ is unique iff for every class $C$ that satisfies (1), we have that: $$x\in C \Rightarrow x\in U$$ ... But here i got stuck, since i don't know how to show this property for $C$, nor construct $C≠U$ that satisfies (1). Any ideas?