Definition:Existential Quantifier/Unique/Definition 1

Definition
There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, :
 * $\exists x : \paren {\map P x \land \forall y : \paren {\map P y \implies x = y} }$

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.

Also see

 * Equivalence of Definitions of Unique Existential Quantifier