Greatest Element/Examples/Finite Subsets of Natural Numbers less Empty Set

Examples of Greatest Elements
Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.

Let $\GG$ denote the set $\FF \setminus \O$, that is, $\FF$ with the empty set excluded.

Consider the ordered set $\struct {\GG, \subseteq}$.

$\struct {\FF, \subseteq}$ has no greatest element.

Proof
$A \in \GG$ is the greatest element of $\struct {\GG, \subseteq}$.

From Greatest Element is Maximal, $A$ is a maximal element of $\struct {\GG, \subseteq}$.

But from Maximal Element: Finite Subsets of Natural Numbers less Empty Set, $\struct {\GG, \subseteq}$ has no maximal element.

Hence $A$ cannot be the greatest element of $\struct {\GG, \subseteq}$.

The result follows from Proof by Contradiction.