Definition:Existential Quantifier/Unique

Definition
The symbol $\exists !$ denotes the existence of a unique object fulfilling a particular condition.


 * $\exists ! x: P \left({x}\right)$

means:
 * There exists exactly one object $x$ such that $P \left({x}\right)$ holds

or:
 * There exists one and only one $x$ such that $P \left({x}\right)$ holds.

Formally:
 * $\exists !x: P \left({x}\right) \dashv \vdash \exists x: P \left({x}\right) \land \forall y: P \left({y}\right) \implies x = y$

In natural language, this means:


 * There exists exactly one $x$ with the property $P$
 * is logically equivalent to:
 * There exists an $x$ such that $x$ has the property $P$, and for every $y$, $y$ has the property $P$ only if $x$ and $y$ are the same object.

The symbol $\exists !$ is a variant of the existential quantifier $\exists$: there exists at least one.

Also denoted as
The symbol $\exists_1$ is also found for the same concept, being an instance of the exact existential quantifier $\exists_n$.