I'm reading Shelah's book "Proper and Improper Forcing" (the first two chapters were recommended for learning the basics of forcing)
Given a quasi-order $P$ we say that $\mathcal{I}$ is a dense subset of $P$ if $$(\forall p \in P) (\exists q \in \mathcal{I}) (p\le q)$$
We say that it is open if for any $p,q \in P$ we have that $p \in \mathcal{I} \wedge p\le q$ then $q \in \mathcal{I}$
$G$ is called directed if every two elements in $G$ has an upper bound in $G$.
$G$ is called downward closed if for every $p \in G$ and $q\in P$ if $q \le p$ then $q \in G$
A subset $G$ of $P$ is called generic over $V$ if it is directed, downward closed and for any dense and open subset of $P$ that is in $V$, the intersection with $G$ is non-empty.
Now they give proof that if $P$ has no trivial branches then a generic set cannot exist in the universe, saying that if $G \in V$ then $P\backslash G \in V$ and it is dense and open, how come? I can't figure that out.