Basic set questions I would really appreciate it if you could explain the set notation here
$$\{n ∈ {\bf N} \mid (n > 1) ∧ (∀x,y ∈ {\bf  N})[(xy = n) ⇒ (x = 1 ∨ y = 1)]\}$$
1) What does $∀x$ mean?
2) I understand that $n ∈ {\bf N} \mid (n > 1) ∧ (∀x,y ∈ {\bf N})$ means $n$ is part of set $\bf N$ such that $(n > 1) ∧ (∀x,y ∈ {\bf N})$. What do the $[\;\;]$ and  $⇒$ mean?
3)  Prove that if $A ⊆ B$ and $B ⊆ C$, then $A ⊆ C$
I could prove it by drawing a Venn diagram but is there a better way?

1) $(\forall x)$ is the universal quantifier. It means "for all $x$".
2) $[ ]$ is the same as a parenthesis. Probably, the author did not want to use too many round parenthesis because it would get too confusing.
$\Rightarrow$ is implies.
3) Suppose $x \in A$. Since $A \subset B$, by definition $x \in B$. Since $B \subset C$, then $x \in C$. So $x \in A$ implies $x \in C$. This is precisely the definition of $A \subset C$. 
