Answer
the elements of the asked proof that there is a unique
elements are Existence and uniqueness.
Work Step by Step
A proof that there is a unique element $x$ such that $P(x)$ consists of two parts existence and uniqueness.
Existence: we prove that there exists an element for which $P(x)$ is true.
Uniqueness: we prove that if there are two elements $y$ and $z$ for which $P(x)$ is true, then $y$ and $z$ have to be equal
hence we can say that the existing element is then unique
