Singleton Class can be Formed from Set

Theorem

Let $V$ be a basic universe.

Let $a \in V$ be a set.

Then the singleton class $\set a$ can be formed, which is a subclass of $V$.

Proof

Using the axiom of specification, let $A$ be the class defined as:

$A := \set {x: x \in V \land x = a}$

That is:

$A = \set a$

By the axiom of extension, $\set a$ is the only such class which has $a$ as an element.

Also see

• Definition:Singleton 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