Singleton Class of Set is Set
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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 $4.1$