Proving the uniqueness of the empty set

While proving some assertion about sets, the need for showing that a set is equals to the empty set may arise. In that situation, a lot of confusion appear if the uniqueness of the empty set is unknown, because showing that the empty set, which we know has no element, is equal to another different, but also with no element, makes no sense, since the empty set is just one, the $\emptyset$.

So, we can claim that there's only one set with no elements.

Proof: Suppose $A, B$ are sets with no elements. It's known that $A \subset B$, and $B \subset A$. Then, by definition of equality, $A = B$.

In this way, in order to show that some set is equals to the empty set, it's necessary only to show that the set has no elements, and, due to the uniqueness, it will be undoubtedly the empty set $\emptyset$.