I believe I've proved that the power semigroup of non-negative integers with addition has a trivial automorphism group. The proof is a bit long, completely elementary and rather unremarkable (as the fact itself probably) so I won't post it here. However, I spent a long time thinking it up. Probably way too long. So I would like to know whether the fact

• is actually a fact;

• is completely trivial;

• follows immediately from a theorem that's maybe not that trivial;

• has any significance.

Added: For a semigroup $(S,\star)$, the power semigroup of $S$, denoted by $P(S)$ is the semigroup of all non-empty subsets of $S$ with the operation $A\star B=\{a\star b\,|\,a\in A,b\in B\}.$

Added later: I'm confused by the answers given so far. The answerers seem to be showing that any automorphism of $P(\Bbb N)$ has to fix every singleton. Showing this was the first step in my proof, but I couldn't see how it showed immediately that such an automorphism has to fix everything. That's why my proof was getting longer and longer. I kept showing that an automorphism had to fix more and more things, until it had to fix all things. If that was unnecessary, I would like to understand it. Even if it is completely obvious, please do spell it out for me if possible. My mathematical spirits haven't been above sea level for a very long time, and being unable to understand something that people seem to consider obvious always brings me down.