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

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

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

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

$\struct {\GG, \subseteq}$ has no smallest element.

Proof
From Minimal Element: Finite Subsets of Natural Numbers less Empty Set, $\struct {\GG, \subseteq}$ has more than one minimal element.

The result follows from Ordered Set with Multiple Minimal Elements has no Smallest Element.