How to express that a set has only one maximum In a given set of numbers N there is one, only one, that is larger than all the others. What is the best way to express this?
My guess:
$\exists x \in N |( x > \forall y \in N \wedge \forall y \in N = x \Rightarrow y = x) $
or:
$\exists ! x \in N | x > \forall y \in N$
Is that correct or do I need to indicate that x is larger than all the elements except itself?
For instance:
$\exists ! x \in N | x > \forall y \in N | y \neq x $
 A: How about
$$ \exists ! x \in N, \forall y \in N, x \neq y \implies x > y$$
A: Writing $x > \forall y \in N$ is highly non-standard. It might be understood if you clarify elsewhere, or if it's clear from context, but you should probably not expect it. Similarly, using your $\mid$ as a sort of reverse implication outside of set notation is also non-standard.
The standard way to write this would be
$$
\exists x \in N(\forall y(x \neq y \to x > y)).
$$
Note that $x > y$ already precludes the possibility that there might be another $x'$ which is also greater than all other elements, so you don't need to express uniqueness separately.
A: Another possible option is "there is a unique element $x$ of $X$ such that $x = \max (X)$"
$$\exists_{1} x \in X: \quad x = \max (X)$$
Note that one usually writes $\forall y: x \geq y$. Also note that the statement
$$\exists x \in X \quad \forall y \in X: x > y$$
means that $x>x$, a constradiction to the axioms of equality relation. Therefore
$$\exists x \in X \quad \forall y \in X/\{x\}: \quad x > y$$
or
$$ \exists x \in X\quad \forall y \in X: \quad x \neq y \implies x > y$$
In two last cases you do not need to state the uniqueness of such $x$ anymore.
