What does the union symbol (∪) before the set builder notation mean? I am reading Robert André's book Axioms and Set Theory and, maybe starting at chapter 12, he uses the notation like $\mathbb{N}=\cup\{n \mid n \in \mathbb{N}\}$, where he puts a union symbol (∪) before the set builder notation. I do not know what it means, though I suspect it is a typo.
Hence, my question: what does $\cup\{x \mid \phi(x)\}$ mean?
 A: In a set theory development of the natural numbers,  what we do is we start with nothing but sets and then build structures within that mimic our knowledge of numbers and operations on them.  The standard construction for the natural numbers is to set
$$0:=\emptyset $$
and then inductively define each following natural number by the successor function $s(x)$ which gives the next number after $x$ as
$$s(x+1)=x\cup \{x\}$$
So   $0=\emptyset$ $1=\{0\}=\{\emptyset \}$,  $2=1 \cup \{1\}=\{0\}\cup \{1\}=\{0,1\}$, etc.
Note in this construction the only objects are the empty set and sets built from the empty set,  the numbers are just names we assign to certain configurations of the sets.   In this configuration,  it is true that any natural number is just the set of all the previous numbers
$$n=\{0,1,2,...,n-1\}$$
Thus it makes the sense to take the union of the numbers because numbers are just names for special sets.   And when you take the union of all of them, you get all natural numbers.
Added: For your exercise, you just want to show the two sets are equal, so as usual, show they are subsets of each other.  So show any natural number is in that union,  and then show that any member of that union is in fact a natural number
Second addendum:   Technically the author should have written
$$\bigcup_{x\in \mathbb N}x$$
but when the context of the index set is obvious, we often do abuse of notation and skip writing the index set and go straight to
$$\bigcup x$$
A: The notation is not mysterious and is not “technically wrong” or “abusive” as others commented. You find it in several books; I learned it in the appendix of Kelley's book “General Topology”.
If $A$ is a set, then
$$
\mathop{\cup}A=\{x\mid \exists y\,(y\in A\land x\in y)\}
$$
For instance, if $A=\{\{a,b\},\{c\}\}$, then $\mathop{\cup}A=\{a,b,c\}$.
The book you link is not very consistent in notation, I should say.
In the “almost formal” set theory that's developed in the book, everything is a set. And the natural numbers are defined recursively by
$$
0=\emptyset,\quad n^+=n\cup\{n\}
$$
and collecting what's obtained in this way in the set $\mathbb{N}$. Let's see what happens when the union is applied to $1=\{0\}$ instead of $\mathbb{N}$:
$$
\mathop{\cup}\{n\mid n\in 1\}=0=\emptyset
$$
With $2=\{0,1\}$ we get
$$
\mathop{\cup}\{n\mid n\in 2\}=1
$$
There seems to be a pattern. And indeed there is: if $n$ is a natural number and $y\in n$, then $y\subset n$. Since, by definition,
$$
0\subset 1\subset 2\subset\dotsb
$$
you have that, when $m=k^+$
$$
\mathop{\cup}\{n\mid n\in m\}=k
$$
which you can prove by induction. It follows that
$$
\mathop{\cup}\{n\mid n\in\mathbb{N}\}=\mathbb{N}
$$

The notation $\mathop{\cup}A$ is justified by a specific axiom of set theory: the axiom of union states that for any $A$ there exists $B$ such that, for every $x\in B$, there is $y\in A$ with $x\in y$ (probably not the same notation of your book). Using specification, we can isolate from that (possibly big) set the one that we need to use and denote it as shown above.
The “binary union” $A\cup B$ is defined to be $\mathop{\cup}\{A,B\}$.
A: It means the union of all such sets which are described by {$x | \phi(x)$}.
A: Your notation is undefined (I think) as you need to state what $x$ ranges over (context dependent). But Robert is not making a typo. You might recognise it more if I rewrite it like this:
$$\Bbb N=\bigcup_{n\in\Bbb N}\{n\}$$
Which is true of all sets. $\cup$ just means union, and sometimes people don’t put [the set or property which indexes the union] on the bottom, but rather to the side, as Robert has done. That’s what “$| \phi(x)$” means: the union is taken over those sets $x$ such that $\phi(x)$. It would be equivalent (provided you had a background from where $x$ will come, like a universe or some other set $A$) to write: $$\cup\{x\in A:\phi(x)\}=\bigcup_{x\in A:\phi(x)}\{x\}$$
