Set theory/Aristotelian Syllogisms/1/Exercise

Prove the following (set-theoretical versions of) syllogisms of Aristotle. Let ${}A,B,C$ denote sets.

1. Modus Barbara: ${}B\subseteq A$ and ${}C\subseteq B$ imply ${}C\subseteq A$.
2. Modus Celarent: ${}B\cap A=\emptyset$ and ${}C\subseteq B$ imply ${}C\cap A=\emptyset$.
3. Modus Darii: ${}B\subseteq A$ and ${}C\cap B\neq \emptyset$ imply ${}C\cap A\neq \emptyset$.
4. Modus Ferio: ${}B\cap A=\emptyset$ and ${}C\cap B\neq \emptyset$ imply ${}C\not \subseteq A$.
5. Modus Baroco: ${}B\subseteq A$ and ${}B\not \subseteq C$ imply ${}A\not \subseteq C$.

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