It is known that Morse-Kelly (MK) set theory forms a metatheory for ZFC. For example:
MK proves Con(ZFC). In fact, Joel David Hamkins claims in his blog post "Kelly-Morse set theory implies Con(ZFC) and much more" that in MK "there is a transitive model of ZFC, and furthermore that the universe $V$ is the union of an elementary chain of elementary rank initial segments $V_\theta$ of $V$, each of which , in particular, is a transitive model of ZFC."
My question is this: If in MK there is a transitive model $M$ of ZFC, is there in MK the generic extension $M$[$G$], where the generic $G$ is definable in MK? Also, if so, is there a $M$[$G$] in MK in which cardinals are collapsed?