Prove that $A$ is a transitive set iff the following holds:$B \in C$ and $C \in A$ then $B \in A$

Following problem is from Pinter’s book of set theory

Prove that $A$ is a transitive set iff the following holds:

if $B\in C$ and $C\in A$,then $B\in A$

$6.5$ Definition A set $A$ is called transitive if, for each $x \in A$, $x\subseteq A$.

Are $B$ and $C$ transitive sets?

If $A^+=A\cup$ {A} could I say $B= A^+$ lexicographically speaking?

Attempted proof

If $A$ is transitive then $x \in A$, $x \subseteq A$. after this I am stuck Suppose that 𝑎

You should be able to get a little further than that. For that direction the most obvious starting point is something like this:

Suppose that $a$ is transitive and that $b\in c\in a$; we want to show that $b\in a$. Since $a$ is transitive and $c\in a$, we know that $c\subseteq a$.

That much should be more or less automatic, and at that point it’s easy to use the hypothesis that $b\in c$ to get the desired conclusion.

And for the other direction:

Suppose that $b\in a$ whenever $b\in c\in a$; we want to show that $a$ is transitive, i.e., that if $c\in a$, then $c\subseteq a$.

Now how can one show that $c$ is a subset of $a$? One of the most straightforward approaches is to show that every member of $c$ is also a member of $a$, and the hypothesis here makes that very easy.