Proof

For any two given non-empty sets $X,Y$ assume, their elements $x\in X,y\in Y$ are sets themselves.
By the extensionality principle they are equal $x=y$ if they have the same elements.
This argument can be repeated for the elements of these elements (if any). We can repeat this argument for all elements of elements of elements, etc.
At some point, we might arrive at elements $x\in X$ and $y\in Y$ which do not contain any elements. Note that the existence of such empty elements $x=\emptyset$ and $y=\emptyset$ is ensured by the axiom of empty set.
Now, the extensionality principle says again, that $x=y$ are equal, even though $x=\emptyset$ and $y=\emptyset$.
In other words, all empty sets are not distinguishable. Thus, $\emptyset$ is unique.
∎

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