Number of Elements of a Set

Let $X$ be a set and let $n\in\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers and $[n]$ denotes the initial segment determined by $n$.

We say that $X$ has $n$ elements if there exists a bijection

By Uniqueness of the Number of Elements there is at most one such $n$; when it exists, it is called the number of elements of $X$ and is written $|X|$.