Empty set as a singleton So, it is obvious that the empty set is a subset of every set, such that ∀A:∅⊆A.

Therefore, if the null set is an element of some set A, such that A = {∅}, is the empty set considered a singleton element of A?

Thank you.
Yes, the set $\{\varnothing\}$ is a set whose only element is the empty set $\varnothing$.
Divorce your mind from any thought that of an element being a set in its own right, from the role of the element as an element in another set.  If $x$ is the only element in $A$ then $x$ is a singleton element of $A$.  It doesn't matter what $x$ is and if $x$ just happens to be a set, its elements have nothing whatsoever to do with what kind of element $x$ is to $A$.

So $A=\{\emptyset\}$.  The $A$ has a single element.  That singleton element is $\emptyset$.  

That's all that needs to be said.

.....

Um... I'm assuming you are taking "singleton" to mean the only element of  a set?