Characterization of Paracompactness in T3 Space/Lemma 5

Theorem
Let $X$ be a set.

Let $\AA$ and $\VV$ be sets of subsets of $X$.

For each $V \in \VV$, let:
 * $V^* = X \setminus \ds \bigcup \set{A \in \AA | A \cap V = \O}$

Then:
 * $\forall V \in \VV: V \subseteq V^*$

Proof
Let $V \in \VV$.

Let $\AA_V = \set{A \in \AA | A \cap V = \O}$.

From Subset of Set Difference iff Disjoint Set:
 * $\forall A \in \AA_V : V \subseteq X \setminus A$

We have: