Finite Subsets form Ideal

Theorem
Let $X$ be a set.

Let $\map {\operatorname {Fin} } X$ be the set of all finite subsets of $X$.

Then $\map {\operatorname {Fin} } X$ is ideal in $\struct {\powerset X, \subseteq}$

where $\powerset X$ denotes the power set of $X$.

Non-Empty
By Empty Set is Subset of All Sets:
 * $\O \subseteq X$ and $\O$ is finite.

By definition of $\operatorname {Fin}$:
 * $\O \in \map {\operatorname {Fin} } X$

Thus by definition:
 * $\map {\operatorname {Fin} } X$ is non-empty.

Directed
This follows from Finite Subsets form Directed Set.

Lower
Let $x \in \map {\operatorname {Fin} } X$, $y \in \powerset X$ such that
 * $y \subseteq x$

By definition of $\operatorname {Fin}$:
 * $x$ is a finite set.

By Subset of Finite Set is Finite:
 * $y$ is a finite set.

By definition of power set:
 * $y \subseteq X$

Thus by definition of $\operatorname {Fin}$:
 * $y \in \map {\operatorname {Fin} } X$

Hence $\map {\operatorname {Fin} } X$ is ideal in $\struct {\powerset X, \subseteq}$