(A cap C) cup (B cap Complement C) = Empty iff B subset C subset Complement A

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Let $A$, $B$ and $C$ be subsets of a universe $\Bbb U$.


$\paren {A \cap C} \cup \paren {B \cap \map \complement C} = \O \iff B \subseteq C \subseteq \map \complement A$

where $\map \complement C$ denotes the complement of $C$ in $\Bbb U$.


\(\ds \paren {A \cap C} \cup \paren {B \cap \map \complement C}\) \(=\) \(\ds \O\)
\(\ds \leadstoandfrom \ \ \) \(\ds A \cap C\) \(=\) \(\ds \O\) Union is Empty iff Sets are Empty
\(\, \ds \land \, \) \(\ds B \cap \map \complement C\) \(=\) \(\ds \O\)
\(\ds \leadstoandfrom \ \ \) \(\ds C\) \(\subseteq\) \(\ds \map \complement A\) Empty Intersection iff Subset of Complement
\(\, \ds \land \, \) \(\ds B\) \(\subseteq\) \(\ds C\) Intersection with Complement is Empty iff Subset
\(\ds \leadstoandfrom \ \ \) \(\ds B\) \(\subseteq\) \(\ds C \subseteq \map \complement A\)