A statement that an object exists and is unique contains two logically separate claims.

Existence

To prove existence, show that at least one object satisfies the required property. This may be constructive—by explicitly producing one—or nonconstructive—by proving one must exist without identifying it directly.

Uniqueness

To prove uniqueness, assume two objects $a$ and $b$ both satisfy the defining property and show that

$$a=b.$$

Existence alone does not imply uniqueness, and uniqueness without existence only says that there cannot be two different solutions.

A theorem stating “there exists a unique $x$ such that $P(x)$” therefore normally has the structure

1. construct or establish at least one solution;
2. show any two solutions must coincide.

Separating the two obligations makes many arguments in algebra, analysis and differential equations much clearer.