Empty Set is Unique/Proof 2

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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:

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

vacuously.


From Axiom of Extension:

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


Hence the result.

$\blacksquare$


Sources