# Is the set of singletons in a Grothendieck universe always equinumerous with the universe itself?

Let $U$ denote a Grothendieck universe, and let $S = \{x \in U | x \mbox{ is a singleton}\}$. Clearly, $|S| \leq |U|$. Is it necessarily true that $|S|=|U|$?

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I'm not sure why $x\mapsto\{x\}$ is not an obvious bijection. –  Asaf Karagila Mar 20 '13 at 0:22
@AsafKaragila Perhaps it is, my intuition about these kinds of things is just terrible. If you're certain you're correct, feel free to post as an answer. –  user18921 Mar 20 '13 at 0:24