Unordered Pairs Exist

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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