# Unordered Pairs Exist

## Contents

## Theorem

Let $A$ and $B$ be classes, which may be either sets or proper classes.

Then:

- $\forall A, B: \left\{{A, B}\right\} \in U$

where $U$ is the universal class.

## Proof

\(\displaystyle \forall A, B\) | \(:\) | \(\displaystyle \exists x: \forall y: \left({y \in x \iff y = A \lor y = B}\right)\) | $\quad$ Axiom of Pairing | $\quad$ | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \forall A, B\) | \(:\) | \(\displaystyle \exists x: x = \left\{ {y: y = A \lor y = B}\right\}\) | $\quad$ Definition of Set Equality | $\quad$ | ||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \forall A, B\) | \(:\) | \(\displaystyle \left\{ {y: y = A \lor y = B}\right\} \in U\) | $\quad$ Element of Universe | $\quad$ | ||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \forall A, B\) | \(:\) | \(\displaystyle \left\{ {A, B}\right\} \in U\) | $\quad$ Definition of Doubleton | $\quad$ |

$\blacksquare$

## Also see

## Sources

- 1963: Willard Van Orman Quine:
*Set Theory and Its Logic*: $\S 7.10$