Unordered Pairs Exist

Theorem

 * $\forall A,B: \{ A,B \} \in U$

Let $A$ and $B$ be classes, either sets or proper classes, in a doubleton (unordered pair). $U$ is the universal class.

Proof
From the axiom of pairing,


 * $\forall A,B: \exists x: \forall y: \left({y \in x \iff y = A \lor y = B}\right)$


 * $\forall A,B: \exists x: x = \{ y | y = A \lor y = B \}$ Definition:Set Equality


 * $\forall A,B: \{ y | y = A \lor y = B \} \in U$ Element of the Universe


 * $\forall A,B: \{ A,B \} \in U$ Definition:Doubleton

Also See

 * Axiom:Axiom of Pairing


 * Definition:Universal Class


 * Definition:Doubleton

Source

 * : $\S 7.10$