Cardinality of Singleton

Theorem
Let $A$ be a set.

Then $\card A = 1$ $\exists a: A = \set a$

where $\card A$ denotes the cardinality of $A$.

Sufficient Condition
Assume that
 * $\card A = 1$

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

where $\sim$ denotes set equivalence.

By Set Equivalence behaves like Equivalence Relation:
 * $\set 0 \sim A$

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

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

Thus

Necessary Condition
Assume that:


 * $\exists a: A = \set a$

Define a mapping $f: A \to \set 0$:
 * $\map f a = 0$

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:
 * $A \sim \set 0 = \N_{< 1}$

Thus by definition of cardinality of finite set:
 * $\card A = 1$