# Definable subsets--theorem from A Shorter Model Theory

So this is a theorem [1] from A Shorter Model Theory and I'm being unable to prove this when it seems like proving this would be quite intuitive and straightforward. Does anyone know a proof?

Let L be the empty signature and A an L-structure so that A is simply a set. Let X be any subset of A, and Y be a subset of dom(A) which is definable in A by a formula of some logic of signature L, using parameters from X. Then Y is either a subset of X, or the complement in dom(A) of a subset of X.

[1] Theorem 2.1.2, page 27.

• You have written $X$ as a subset of $X$. Could you please edit it? – Kyle Dec 29 '14 at 8:45

When $L=\varnothing$ every bijection is an automorphism. If $Y$ is definable over $X$, then every bijection that fixes $X$ pointwise fixes $Y$ setwise. This can only occur if $Y\subseteq X$ or $A\smallsetminus Y\subseteq X$.
• You probably meant to write that $Y$ contains the complement in the latter case. – tomasz Dec 30 '14 at 0:56