Proving via axioms, that for given set, $A$, exist a set $\{(X,Y): X\subset Y \subset A \}$ Can somebody show me how to prove, the statement in the title?

For given set, $A$, exist a set $\{(X,Y): X\subset Y\subset A \}$

Thanks!

To show existence of a set $\{(X,Y): X\subset Y\subset A \}$ for an arbitrary set you can apply the Axiom schema of specification which says that
"Every subclass of a set that is defined by a predicate is itself a set." To do this you would like to replace $\phi$ and $A'$ in the following schema:
$$ \forall w_1 , \dots, w_n \forall A' \exists B \forall x ( x \in B \iff [x \in A' \land \phi (x,w_1, \dots , w_n, A')])$$
Let $A'$ in the formula be the set of all pairs $(X,Y)$ with $X, Y$ sets and let $\phi (x, A') = (x=(X,Y)) \land (X \subset Y \subset A)$. Then $B$ is the set $\{(X,Y): X\subset Y\subset A \}$.
