On Rasiowa Sikorski lemma The Rasiowa Sikorski lemma: 
Let $(P, \le)$ be a poset and $p \in P$. If $D$ is a countable family of dense subsets of $P$ then there exists a $D$-generic filter $F$ in $P$ such that $p \in F$.
Where does $F$ exists? by construction $F \subseteq P$ . But "in $P$" is "$F \in P$"? 
 A: By a filter in $P$ (or, rather, in $\langle P , \leq \rangle$)  we mean a subset $F \subseteq P$ satisfying certain properties (if $p \leq q$ and $p \in F$ then $q \in F$; if $p , q \in F$ then there is an $r \in F$ with $r \leq p,q$).
Unfortunate terminology (especially when you leave out mention of the partial ordering), but I think we're stuck with it at this point.
A: The filter exists in the universe. That is to say that if $V$ is the universe of sets, $P$ is a countable poset and $D_n$ for $n\in\omega$ is a family of dense subsets of $P$, then there is some $F\in V$ which is $D$-generic.
This is often used in forcing, we take $M$ a countable transitive model of $\sf ZFC$, and we take a partial order $P\in M$, since $M$ is countable it only knows about countably many subsets of $P$. So the collection of "all the dense subsets of $P$" relative to $M$ is a countable collection.
Therefore in $V$ itself there is a $P$-generic filter. In the nontrivial case, this generic filter does not exist in $M$, and then we add it to create $M[G]$ which is another countable transitive model of $\sf ZFC$.
