Doubleton of Sets can be Derived using Axiom of Abstraction

Theorem
Let $a$ and $b$ be sets.

By application of the, the set $\set {a, b}$ can be formed.

Hence the doubleton $\set {a, b}$ can be derived as a valid object in Frege set theory.

Proof
Let $P$ be the property defined as:


 * $\forall x: \map P x := \paren {x = a \lor x = b}$

where $\lor$ is the disjunction operator.

Hence, using the, we form the set:
 * $\set {a, b} := \set {x: x = a \lor x = b}$