# Equivalence of Formulations of Axiom of Pairing for Classes

## Theorem

The following formulations of the **Axiom of Pairing** in the context of **class theory** are equivalent:

### Formulation 1

Let $a$ and $b$ be sets.

Then the class $\set {a, b}$ is likewise a set.

### Formulation 2

For any two sets, there exists a set to which those two sets are elements:

- $\forall a: \forall b: \exists c: \forall z: \paren {z = a \lor z = b \implies z \in c}$

## Proof

It is assumed that all classes are subclasses of a basic universe $V$.

### $(1)$ implies $(2)$

Let formulation $1$ of the Axiom of Pairing be assumed:

Let $a$ and $b$ be sets.

Then the class $\set {a, b}$ is likewise a set.

Thus we have that $c = \set {a, b}$ is a set such that both $a \in c$ and $b \in c$.

Thus formulation $2$ of the Axiom of Pairing is seen to hold.

$\Box$

### $(2)$ implies $(1)$

Let formulation $2$ of the axiom of pairing be assumed:

For any two sets, there exists a set to which those two sets are elements:

- $\forall a: \forall b: \exists c: \forall z: \paren {z = a \lor z = b \implies z \in c}$

Then the class $\set {a, b}$ is a subclass of $c$.

We have by hypothesis that $c$ is a subclass of a basic universe $V$.

Hence by the Axiom of Swelledness, every subclass of $c$ is a set.

That is, $\set {a, b}$ is a set.

Thus formulation $1$ of the Axiom of Pairing is seen to hold.

$\blacksquare$

## 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 $4$