Definable sets à la Jech Jech in Set Theory, p. 175 defines definable sets over a given model $(M,\in)$ (where $M$ is a set) as those sets (= subsets of $M$) $X$ with a formula $\phi$ in the set of formulas of the language $\lbrace \in \rbrace$ and some $a_1,\dots,a_n \in M$ such that 
$$ X = \lbrace x \in M : (M,\in) \models \phi[x,a_1,\dots, a_n]\rbrace$$
Jech defines
$$ \text{def}(M) := \lbrace X \subset M : X\ \text{is definable over }\ (M,\in)\rbrace $$
So far, everything is clear to me.
But then, Jech claims:

Clearly, $M \in \text{def}(M)$ and $M \subset \text{def}(M) \subset \text{P}(M)$.

It is clear to me that and why $M \in \text{def}(M)$ and that and why $\text{def}(M) \subset \text{P}(M)$. But I cannot see at once that and why $M \subset \text{def}(M)$ for arbitrary sets $M$.
Is this a typo in Jech's Set Theory or did I misunderstand something?

If $M$ is transitive and $a \in M$ then $a \subseteq M$ and moreover $$a = \{x \in M : (M,{\in}) \vDash x \in a\}.$$ So $a \in \operatorname{def}(M)$.
