Unordered Pairs Exist

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Theorem

Let $A$ and $B$ be classes.


Then:

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

where $U$ is the universal class.


Proof

\(\ds \forall A, B: \exists x: \forall y: \, \) \(\ds \leftparen {y \in x}\) \(\iff\) \(\ds \rightparen {y = A \lor y = B}\) Axiom of Pairing
\(\ds \leadsto \ \ \) \(\ds \forall A, B: \exists x: \, \) \(\ds x\) \(=\) \(\ds \set {y: y = A \lor y = B}\) Definition of Set Equality
\(\ds \leadsto \ \ \) \(\ds \forall A, B: \, \) \(\ds \set {y: y = A \lor y = B}\) \(\in\) \(\ds U\) Element of Universe
\(\ds \leadsto \ \ \) \(\ds \forall A, B: \, \) \(\ds \set {A, B}\) \(\in\) \(\ds U\) Definition of Doubleton

$\blacksquare$


Also see


Sources