Prove whether $A \in B \wedge B \subseteq C \rightarrow A \in C$ true or not If you have three sets A, B, and C; how can you prove that the problem below is true, or give a counterexample to show it's false?
$A \in B \wedge B \subseteq C \rightarrow A \in C$
 A: $A \in B \wedge B \subseteq C \rightarrow$ A is an element in B and every element b in B is also in C $\rightarrow A \in C$.
Not very much, I know, but it's simply the definition of containment.
EDIT: If you want the longer and "mushy" proof, here's a sketch:
Assume $A \in B \wedge B \subseteq C$. So for every element b $\in$ C, implies that b $\in$ C. According to assumption, $A$ is also an element in B, so from about we get A $\in$ C. QED
A: The definition of $\subseteq$ might take a form like
$$X \subseteq Y \quad\iff\quad \forall \alpha.\ \alpha \in X \to \alpha \in Y.$$
Given that, you can rewrite your question
$$A \in B \wedge B \subseteq C \rightarrow A \in C$$
two ways:
$$\alpha \in B \land (\forall \beta.\ \beta \in B \to \beta \in C) \rightarrow \alpha \in C$$
which is true by instantiating $\beta$ with $\alpha$, or
$$\{\alpha\} \subseteq B \wedge B \subseteq C \rightarrow \{\alpha\} \subseteq C$$
which is true by transitivity of $\subseteq$ (if you have this property available).
I hope this helps $\ddot\smile$
A: It is worth adding, not another answer (DanielY's will do fine), but a much more general point.
Often, at the beginning of a maths course, the course instructor (or the exercises in an elementary textbook) will throw in questions like this which are indeed trivial. They want you to spot that such questions are trivial, and thereby reveal that you have a clear and confident grasp of the basic definitions in the area. 
It isn't exactly playing a trick on you, putting such a question into a list of more substantial ones, to see if you are awake and spot that the question is trivial: having done this sort of thing a lot myself over the years, I prefer to think of it as running a reality check, to confirm you are on top of the basics.
Moral: don't be fazed, early on in a course, by questions that look "too easy": they may indeed just be elementary reality checks!
