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?

*

*it means "for all $x$"

*the right arrow means "imply", if $A$ imply $B$ it means that whenever $A$ is true $B$ is also true (note: when $A$ is false $B$ can be either true or false)

*We can prove this very easily: Let $a$ be an element in $A$ then since $A$ is contained in $B$ it means (by definition) that $a$ is an element of $B$. since $B$ is contained in $C$ it means (by definition) that every element of $B$ is in $C$ - in particular $a$ that we know is in $B$ is also in $C$ and so by definition (since evry element of $A$ is an element of $C$) we have that $A$ is contained in $C$
