Definition:Existential Quantifier/Unique/Definition 1

Definition

There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, if and only if:

$\exists x : \paren {\map P x \land \forall y : \paren {\map P y \implies x = y} }$

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 denoted as

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

Some sources, for example 1972: Patrick Suppes: Axiomatic Set Theory, use $\operatorname E !$, which is idiosyncratic, considering the use in the same source of $\exists$ for the general existential quantifier.

Also see

• Equivalence of Definitions of Unique Existential Quantifier

• Results about the unique existential quantifier can be found here.

Sources

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