I'm having trouble with an exercise from Kunen's Introduction to Independence Proofs. This is Exercise F2 from Ch VII.
If $\mathbb{P} \in M$ and $c.c.(\mathbb{P}) < \omega$, show that every $G$ which is $\mathbb{P}$-generic over $M$ is in $M$. Do likewise when ($c.c.(\mathbb{P}) = \kappa \land \mathbb{P}$ is $\kappa$-closed)$^M$.
The book is pretty old, so I'm not sure whether or not the notation above is still used. In any case, here are some relevant definitions.
$c.c.(\mathbb{P})$ is the least $\theta$ such that $\mathbb{P}$ has the $\theta$-c.c.
$\mathbb{P}$ has the "$\theta$-chain condition" ($\theta$-c.c.) iff every antichain in $\mathbb{P}$ has cardinality $<\theta$
A p.o. $\mathbb{P}$ is $\lambda$-closed iff whenever $\gamma < \lambda$ and $\{p_\xi: \xi < \lambda \}$ is a decreasing sequence of elements of $\mathbb{P}$ (i.e., $\xi < \eta \rightarrow p_\xi \geq p_\eta$), then $$\exists q \in \mathbb{P} \forall \xi < \gamma (q \leq p_\xi)$$
Any form of help (whether it be a full solution or a hint) is appreciated
This is one of those things where a proof by contrapositive is much neater. Let me outline the steps, and you can fill in the details. First, let me assume that $\Bbb P$ is separative, the claims still work otherwise, but they require a bit more fiddling to it.
If $\Bbb P$ is $\kappa$-closed, then there is a sequence of descending conditions of length $\kappa$, or the set of atoms (=minimal conditions) is dense.
Let $\{p_\alpha\mid\alpha<\kappa\}$ be a descending sequence, then there is an antichain of size $\kappa$.
Therefore, if $\Bbb P$ is $\kappa$-closed and has the $\kappa$-c.c., then the set of atoms is dense, which means that a generic filter is fully determined by choosing a single atom, and therefore the generics are all in $V$.