Empty Set can be Derived from Comprehension Principle

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

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

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

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

Hence the result by definition of the empty set.