Empty Set is Unique/Proof 2

Theorem

The empty set is unique.

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:

$x \in A \iff x \in B$

From Axiom of Extension:

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

Hence the result.

$\blacksquare$