Empty Set is Unique

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The empty set is unique.

Proof 1

Let $\O$ and $\O'$ both be empty sets.

From Empty Set is Subset of All Sets, $\O \subseteq \O'$, because $\O$ is empty.

Likewise, we have $\O' \subseteq \O$, since $\O'$ is empty.

Together, by the definition of set equality, this implies that $\O = \O'$.

Thus there is only one empty set.


Proof 2

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


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


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


$x \notin A \iff x \notin B$

and so:

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


From the Axiom of Extension:

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

Hence the result.


Proof 3

From Axiom of the Empty Set in the context of class theory, the empty class is a set.

The result follows from Empty Class Exists and is Unique.