Empty Set can be Derived from Axiom of Abstraction

Theorem
The empty set can be formed by application of the.

Hence the empty set can be derived as a valid object in Frege set theory.

Proof
Let $P$ be the property defined as:


 * $\forall x: \map P x := \neg \paren {x = x}$

Hence, using the, we form the set:
 * $\O := \set {x: \neg \paren {x = x} }$

where the property ${\map P x}$ is:
 * $\neg \paren {x = x}$

Since we have that:
 * $\forall x: x = x$

it is seen that $\O$ as defined here has no elements.

By definition of Frege set theory, given any property $P$, there exists a unique set which consists of all and only those objects which have property $P$:


 * $\set {x: \map P x}$

Therefore, there exists a unique set
 * $\O := \set {x: \neg \paren {x = x} }$

which the property $P= \neg \paren {x = x}$, which consists of no elements.

Hence, by the definition of the empty set, this set is a valid object in Frege set theory.

Hence the result by definition of the empty set.