Doubleton Class can be Formed from Two Sets

Theorem

Let $V$ be a basic universe.

Let $a, b \in V$ be sets.

Then the doubleton class $\set {a, b}$ can be formed, which is a subclass of $V$.

Proof

Using the Axiom of Specification, let $A$ be the class defined as:

$A := \set {x: x \in V \land \paren {x = a \lor x = b} }$

That is:

$A = \set {a, b}$

By the Axiom of Extension, $\set {a, b}$ is the only such class which has $a$ and $b$ as elements.

Also see

• Definition:Doubleton Class

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 4$ The pairing axiom