Rice-Shapiro theorem

(Rice's theorem for r.e. index sets) For any finite set $D=\left\{ x_{1}, \ldots , x_{n}\right\}$, we say $\left[x_{1}, \ldots , x_{n}\right]$ (in any order) is a code of $D$. Show that an index set $A$ is r.e. if and only if

1. $x \in A$ and $W_{x} \subseteq W_{y}$ imply $y \in A$;

2. $x \in A$ implies that there exists $y \in A$ such that $W_{y}$ is a finite subset of $W_{x}$; and

3. There exists an r.e. set $B$ that contains codes of all and only finite sets $W_{x}$ such that $x \in A$ (i.e., for each $x \in A$ such that $W_{x}$ is finite, $B$ contains at least one of its code, and for every $\left[x_{1}, \ldots , x_{n}\right] \in B$ and every $x$ such that $\left.W_{x}=\left\{ x_{1}, \ldots , x_{n}\right\} , x \in A\right)$.