Singleton is Finite

Theorem
Let $x$ be arbitrary.

Then $\left\{ {x}\right\}$ is a finite set.

Proof
Define a mapping $f: \left\{ {x}\right\} \to \N_{< 1}$:
 * $f\left({x}\right) = 0$

By definition of singleton:
 * $\forall y, z \in \left\{ {x}\right\}: f\left({y}\right) = f\left({z}\right) \implies y = z$

By definition:
 * $f$ is an injection.

By definition of initial segment of natural numbers:
 * $\N_{< 1} = \left\{ {0}\right\}$

By definition of $f$:
 * $\forall n \in N_{< 1}: \exists z \in \left\{ {x}\right\}: f\left({z}\right) = n$

By definition:
 * $f$ is a surjection.

By definition:
 * $f$ is a bijection.

By definition of set equivalence:
 * $\left\{ {x}\right\} \sim \N_{< 1}$

Thus by definition:
 * $\left\{ {x}\right\}$ is a finite set.