Defining a set using a predicate I encountered this notation in a proof and I'm a bit confused on the message it's trying to convey:

$ B = \{ n \in \mathbb N \mid \neg P(n) \} $

Here's the notation itself within the greater context of the proof (or proof 'fragment' more like):
...
$ P(n): $ predicate-definition
Assume $ \exists n \in \mathbb N, \neg P(n) $
Then $ B = \{ n \in \mathbb N \mid \neg P(n) \} \;\;\;\;\;\;\;\; $ #Since $ B $ is non empty and a subset of $ \mathbb N $ it must have a least element
Then $ \forall b \in B, \Rightarrow \neg P(b) $
Then $ \exists c \in B, \forall d \in B, c \leq d, \Rightarrow c - 1 \notin B \Rightarrow P(c - 1) $
...
My question is, what exactly is $ B $ ? Is it the set of all the natural numbers that make $ P(n) $ false? 

That is exactly right. The notation $X=\{x\ |\ y\}$ means that $X$ is the set of all $x$ such that condition (or conditions) $y$ are satisfied. $x$ and $y$ can be more complex statements, as is the case in your example
