In a paper by B. Mitchell, I stumbled into the following sentence:

"In the summer of 1986 Woodin discovered the second of the forcing orders associated with a Woodin cardinal, the extender algebra. This forcing goes back to the class forcing of Vopěnka [Vopěnka and Hájek, 1972], by which any set is generic, by a class forcing, over any given class model of set theory."

If I interpreted the result correctly, it means that for every inner model $M$ and for every set $x\in V$, there is a class forcing notion $\mathbb{P}$, definable over $M$, s.t. $x$ is $\mathbb{P}$-generic over $M$.

I looked up the reference, which was the book "The theory of semisets", but it was really hard to figure out anything because of the uncommon symbolization.

My questions are the following:

1) Is my formulation of this result correct?

2) Is there another, more approachable, reference where I could find its proof (and maybe more information on class forcing)?

Fine Structure and Class Forcing(Walter de Gruyter, 2000) - books.google.com/books/about/… – François G. Dorais♦ Jul 10 '12 at 21:12