Let $M$ be a structure in the empty language (i.e. no constants, no predicates, no functions). Suppose $A\subset M$. I would like to show that $A$ is definable in $M$ with parameters if and only if $A$ is finite or $M\setminus A$ is finite.
I tried to use the fact that if $A$ is definable over some parameters, any automorphism of $M$ fixing all the parameters preserves membership in $A$. I'm not sure how to proceed and I would appreciate some help.
(My thought so far:
If $A$ is definable over $B=\{b_1,...,b_n\}$, then since any bijection $e:M\rightarrow M$ is an automorphism, if $x\in A\setminus B$ and $y\in M\setminus B$ then a bijection that just transposes $x$ and $y$ is an automorphism fixing $B$ so $x\in A$ iff $y\in A$. Thus, the only elements potentially not in $A$ are the ones in $B$, which is finite. But then it means $M\setminus A$ must always be finite. Is this correct?)