Find a set such that $A\in B$ and $A\subseteq B$

Question

Find a set such that $A\in B$ and $A\subseteq B$.

I was thinking about the $\mathcal P(A)$ to be the answer of this but I'm not sure about $A\subseteq B$.

Answer 1

Let $A=\{x\}$ and $B=\{\{x\}, x\}$

Then, $A\in B$ and $A\subseteq B$; since $\forall y\in A$, $y \in B$

$P(A)= \{x| x\subseteq A \}$

Suppose $A=\{x\}$, then $P(A)=B=\{\{x\}, \{\emptyset \}, \{x, \emptyset \}\}$

As you can see, $A\in P(A)$, but $A\not \subseteq P(A)$. Since, $x\notin P(A)$

Answer 2

Every ordinal and its successor as defined by Von Neumann satisfy this property.

See Von Neumann definition of ordinals