Empty Set is Unique/Proof 2

Proof
Let $A$ and $B$ both be empty sets.

Thus:


 * $\forall x: \neg \paren {x \in A}$

and:
 * $\forall x: \neg \paren {x \in B}$

Hence:
 * $x \notin A \iff x \notin B$

and so:
 * $x \in A \iff x \in B$

vacuously.

From the Axiom of Extension:


 * $\forall x: \paren {x \in A \iff x \in B} \iff A = B$

Hence the result.