Doubleton of Sets can be Derived using Comprehension Principle

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Theorem

Let $a$ and $b$ be sets.

By application of the comprehension principle, 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 we form the set:

$\set {a, b} := \set {x: x = a \lor x = b}$

$\blacksquare$


Sources