Collection of Sets Equivalent to Set Containing Empty Set is Proper Class

Theorem

Let $S = \set \O$ be the singleton whose element is the empty set.

Let $C$ be the collection of all sets which are equivalent to $S$.

$C$ is a proper class.

Proof

By definition of cardinality, $C$ is the collection of all singletons:

$\set {x: \exists y: x = \set y}$

Define a class mapping $f: C \to V$, where $V$ is the universal class, such that $\map f {\set x} = x$.

This is a mapping on the domain $C$, as all elements of $C$ are singletons.

Take an arbitrary $y \in V$.

Then by definition of $C$:

$\set y \in C$

and by definition of $f$:

$\map f {\set y} = y$

Thus:

$\exists x \in A: \map f x = y$

and $f$ is a surjection.

From Universal Class is Proper, $V$ is proper.

It follows from Surjection from Class to Proper Class that $C$ is proper.

$\blacksquare$

Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $3$: Cardinality: Exercise $2$