After working with nested quantifiers and their negations, we now refine existential quantification. Unique existence says not only that an element exists, but that exactly one element exists. This idea appears in inverses, identities, functions, coordinates, and solutions of equations. In this lesson, unique existence will be expressed using $\exists!$, expanded into ordinary quantifiers, and tested through examples involving quantifier order and universe dependence.

The symbol $\exists!$ is called the \textbf{unique existence quantifier}. The statement $$\begin{equation*} \exists!x\,p(x) \end{equation*}$$ is read as “there exists a unique $x$ such that $p(x)$.” It means that at least one element satisfies $p(x)$ and no two distinct elements satisfy $p(x)$.