Finite Class is Set

Theorem
Let $A$ be a finite class.

Then $A$ is a set.

Proof
Let it be assumed that all classes are subclasses of a basic universe $V$.

The proof proceeds by induction.

For all $n \in \N$, let $\map P n$ be the proposition:
 * If $A$ is a finite class with $n$ elements, then $A$ is a set.

The axiom of the empty set gives that the empty class $\O$ is a set.

From Empty Set is Unique, $\O$ is the only set (and hence class) with $0$ elements.

Thus $\map P 0$ is seen to hold.

Basis for the Induction
Let $A$ be a singleton class.

Thus by definition it has $1$ element, which we will call $x$.

$x$ is a class containing $1$ or more elements, one of which we may call $y$.

We have that $A$ is a subclass of $V$.

As $V$ is a basic universe, the axiom of transitivity holds.

Hence:
 * $y \in x \land x \in A \implies y \in A$

But then $x \in A$ and $y \in A$ where $x \ne y$.

This contradicts our assertion that $A$ is a singleton class.

Hence:
 * $x = \O$

It has been established that $\O$ is a set.

Hence $A$ is a class whose $1$ element is a set.

From Singleton Class of Set is Set, it follows that $A$ is a set.

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:
 * If $A$ is a finite class with $k$ elements, then $A$ is a set.

from which it is to be shown that:
 * If $A$ is a finite class with $k + 1$ elements, then $A$ is a set.

Induction Step
This is the induction step:

Let $A$ have $k + 1$ elements.

Let each of those elements be assigned a label:
 * $a_1, a_2, \ldots, a_{k + 1}$

according to a bijection $\phi: A \to k$:
 * $\map \phi {a_k} = k - 1$

This bijection $\phi$ is guaranteed to exist by definition of finite class.

Consider $A$ as the union of the classes:
 * $\set {a_1, a_2, \ldots, a_k} \cup \set {a_{k + 1} }$

By the basis for the induction:
 * $\set {a_{k + 1} }$ is a set.

By the induction hypothesis:
 * $\set {a_1, a_2, \ldots, a_k}$ is a set.

By the axiom of unions:
 * $\set {a_1, a_2, \ldots, a_k} \cup \set {a_{k + 1} }$ is a set.

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore, for all $\forall n \in \N$:
 * If $A$ is a finite class with $n$ elements, then $A$ is a set.

That is, all finite classes are sets.