Singleton Class of Set is Set

Theorem

Let $x$ be a set.

Then the singleton class $\set x$ is likewise a set.


Proof

Let $x$ and $y$ be sets.

Let $x = y$.

From Doubleton Class of Equal Sets is Singleton Class, the doubleton class $\set {x, y}$ is the singleton class $\set x$.

From the axiom of pairing, the doubleton class $\set {x, y}$ is a set when $x$ and $y$ are sets.

Hence $\set x$ is a set.

$\blacksquare$


Sources

  • 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 4$ The pairing axiom: Corollary $4.1$
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