Doubleton of Sets can be Derived using Axiom of Abstraction

Theorem

Let $a$ and $b$ be sets.

By application of the Axiom of Abstraction, 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 Axiom of Abstraction, we form the set:

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

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 7$ Frege set theory