I'm studying a subject on axiomatic set theory (Zermelo-Fraenkel). One of the books we use is Elements of set theory by Herbert Enderton.
Enderton (and the notes) define for any set $a$, the successor $a^+$ by \begin{equation} a^+ = a \cup \lbrace a \rbrace \end{equation} Now they say that a set $A$ is transitive iff: \begin{align} \forall a \forall x (x \in a \wedge a \in A \rightarrow x\in A) \end{align} Now i have to prove (exercise given by a teacher): If $A$ is a transitive set, then $A \in A^+$ and $A \subseteq A^+$.
My question, it is necessary that $A$ be a transitive set? I mean it is not obvious that always $A \in A^+$ and $A \subseteq A^+$? Because in another book use this fact to prove a lot of things (Patrick Suppes).
Thanks and sorry for my poor English.