Way Below in Lattice of Power Set

Theorem
Let $X$ be a set.

Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \preceq}\right)$ be a lattice of power set of $X$ where $\mathord\preceq = \mathord\subseteq \cap \left({\mathcal P\left({X}\right) \times \mathcal P\left({X}\right)}\right)$

Let $x, y \in \mathcal P\left({X}\right)$.

Then $x \ll y$
 * for every a set $Y$ of subsets of $X$ such that $y \subseteq \bigcup Y$
 * then there exists a finite subset $Z$ of $Y$: $x \subseteq \bigcup Z$

where $\ll$ denotes the way below relation.

Sufficient Condition
Let $x \ll y$

Let $Y$ be a set of subsets of $X$ such that
 * $y \subseteq \bigcup Y$

By definitions of power set and subset:
 * $Y \subseteq \mathcal P\left({X}\right)$

By the proof of Power Set is Complete Lattice:
 * $\bigcup Y = \sup Y$

By definition of $\preceq$:
 * $y \preceq \sup Y$

By Way Below in Complete Lattice:
 * there exists a finite subset $Z$ of $Y$: $x \preceq \sup Z$

By the proof of Power Set is Complete Lattice:
 * $\bigcup Z = \sup Z$

Thus by definition of $\preceq$:
 * there exists a finite subset $Z$ of $Y$: $x \subseteq \bigcup Z$

Necessary Condition
Suppose
 * for every a set $Y$ of subsets of $X$ such that $y \subseteq \bigcup Y$
 * then there exists a finite subset $Z$ of $Y$: $x \subseteq \bigcup Z$

We will prove that
 * for every a subset $Y$ of $\mathcal P\left({X}\right)$ such that $y \preceq \sup Y$
 * then there exists a finite subset $Z$ of $Y$: $x \preceq \sup Z$

Let $Y$ be a subset of $\mathcal P\left({X}\right)$ such that
 * $y \preceq \sup Y$

By definition of power set:
 * $Y$ is a set of subsets of $X$.

By the proof of Power Set is Complete Lattice:
 * $\bigcup Y = \sup Y$

By definition of $\preceq$:
 * $y \subseteq \bigcup Y$

By assumption:
 * there exists a finite subset $Z$ of $Y$: $x \subseteq \bigcup Z$

By the proof of Power Set is Complete Lattice:
 * $\bigcup Z = \sup Z$

Thus by definition of $\preceq$:
 * there exists a finite subset $Z$ of $Y$: $x \preceq \sup Z$

Thus by Way Below in Complete Lattice:
 * $x \ll y$