Singleton Class of Empty Set is Supercomplete

Theorem

Let $\O$ denote the empty set.

Then the singleton $\set \O$ is supercomplete.

Proof

Let $x \in \set \O$ be any element of $\set \O$.

Then it has to be the case that $x = \O$.

Then every element of $\O$ is an element of $\set \O$ vacuously.

That is, $\set \O$ is swelled.

There is one element of $\set \O$, and that is $\O$.

This is a subclass of $\set \O$.

That is, $\set \O$ is transitive.

The result follows by definition of supercomplete class.

$\blacksquare$

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: Note $2$.