Empty Intersection iff Subset of Complement/Corollary/Proof 1

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Corollary to Empty Intersection iff Subset of Complement

Let $A, B, S$ be sets such that $A, B \subseteq S$.

Then:

$\exists X \in \powerset S: \paren {A \cap X} \cup \paren {B \cap \complement_S \paren X} = \O \iff A \cap B = \O$

where $\overline X$ denotes the relative complement of $X$ in $S$.


Proof

Let there exist such a set $X$.

Then:

\(\displaystyle \) \(\) \(\displaystyle \paren {A \cap X} \cup \paren {B \cap \complement_S \paren X} = \O\)
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle A \cap X = \O \land B \cap \complement_S \paren X = \O\) Union is Empty iff Sets are Empty
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle A \subseteq \complement_S \paren X \land B \subseteq X\) Empty Intersection iff Subset of Complement
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle A \cap B = \O\)

$\blacksquare$