Singleton Classes are Equal iff Sets are Equal

Theorem

Let $a$ and $b$ be sets.

Let $\set a$ and $\set b$ denote the singleton classes of $a$ and $b$.

Then:

$\set a = \set b \iff a = b$

Proof

Let $a = b$.

Then $\set a$ and $\set b$ contain the same elements.

Hence by the axiom of extension it follows that $\set a = \set b$.

Let $\set a = \set b$.

We have that $a \in \set a$.

As $\set a = \set b$ we also have that $a \in \set b$.

But $b$ is the only element of $\set b$.

Hence it must be the case that $a = b$.

$\blacksquare$

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: Note $3$.