Unordered Pairs Exist

 It has been suggested that this page or section be merged into Axiom:Axiom of Pairing. (Discuss)

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 $\quad$

$\blacksquare$