Definition:Existential Quantifier/Unique

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


 * $\exists ! x: \map P x$

means:
 * There exists exactly one object $x$ such that $\map P x$ holds

or:
 * There exists one and only one $x$ such that $\map P x$ holds.

Formally:
 * $\exists ! x: \map P x \dashv \vdash \exists x: \map P x \land \forall y: \map P y \implies x = y$

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.