Consider a ctm $\mathfrak{M}$ of set theory.

Let $X$ be a projectively definable collection of sets of reals (please see my earlier question "Projectively definable family of sets of reals").

My question is: is the type of nondefinable elements in $X$ is definable over $Th(\mathfrak{M})$ or not.

(I assume that $X$ is infinite.)

PS: note that the type of nondefinable elements of $X$ is the type containing all statements of the form: $$ \varphi(x) \rightarrow (\exists y \neq x \varphi(y)),$$ for every formula $\varphi(x)$ plus the formula defining $X$.