Existence of Singleton Set

Theorem

Let $a$ be a set.

Then the singleton set $\set a$ may be constructed such that:

$a \in \set a$

Proof

Let $a$ be a set.

From the Axiom of Pairing the set $\set {a, a}$ may be formed.

From the Axiom of Extension it follows that:

$\set {a, a} = \set a$

$\blacksquare$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 3$: Unordered Pairs
• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
• 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Pairing