Doubleton Class of Equal Sets is Singleton Class

Theorem

Let $V$ be a basic universe.

Let $a, b \in V$ be sets.

Consider the doubleton class $\set {a, b}$.

Let $a = b$.

Then:

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

where $\set a$ denotes the singleton class of $a$.

Proof

Let $A = \set {a, b}$

The existence of $A$ is shown in Doubleton Class can be Formed from Two Sets:

$A := \set {x: \paren {x = a \lor x = b} }$

Let $a = b$.

Then:

\(\ds A\)

\(=\)

\(\ds \set {a, b}\)

Definition of $A$

\(\ds \)

\(=\)

\(\ds \set {a, a}\)

as $a = b$

\(\ds \leadsto \ \ \)

\(\ds x \in \set {a, a}\)

\(\iff\)

\(\ds \paren {x = a \lor x = a}\)

Definition of $A$

\(\ds \leadsto \ \ \)

\(\ds x \in \set {a, a}\)

\(\iff\)

\(\ds x = a\)

Rule of Idempotence

\(\ds \leadsto \ \ \)

\(\ds x \in \set {a, a}\)

\(\iff\)

\(\ds x \in \set a\)

Definition of Singleton Class

$\blacksquare$

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