Cardinality of Singleton

Theorem
Let $A$ be a set.

Then $\left\vert{A}\right\vert = 1$ $\exists a: A = \left\{ {a}\right\}$

where $\left\vert{A}\right\vert$ denotes the cardinality of $A$.

Sufficient Condition
Assume that
 * $\left\vert{A}\right\vert = 1$

By definition of cardinality of finite set:
 * $A \sim \N_{< 1} = \left\{ {0}\right\}$

where $\sim$ denotes set equivalence.

By Set Equivalence is Equivalence Relation:
 * $\left\{ {0}\right\} \sim A$

By definition of set equivalence there exists a bijection:
 * $f: \left\{ {0}\right\} \to A$

By definition of bijection:
 * $f$ is s surjection.

Thus

Necessary Condition
Assume that
 * $\exists a: A = \left\{ {a}\right\}$

Define a mapping $f: A \to \left\{ {0}\right\}$:
 * $F\left({a}\right) = 0$

It is easy to see by definition that
 * $f$ is an injection and a surjection.

By definition
 * $f$ is bijection.

By definition of set equivalence:
 * $A \sim \left\{ {0}\right\} = \N_{< 1}$

Thus by definition of cardinality of finite set:
 * $\left\vert{A}\right\vert = 1$