To this question in Kunen's introduction to independence proofs
on the page 379, in the proof of lemma V.7.3 you never mention $q$ in the proof, only $p$; why? I cannot make sense of this simple proof and lemma.
I've received this answer:
The claim is that there is a q extending p and a countable subset B of kappa such that q forces a certain property of B.
The proof says that if there is no such q and B, then p forces that in the generic extension, there is no countable ground-model subset B of kappa satisfying this property.
And this last statement (which doesn't mention q) leads to a contradiction.
BUT I still do not understand why we can get rid off q and say
then p forces that in the generic extension, there is no countable ground-model subset B of kappa satisfying this property.
I may write down the details for those who do not have the book opened on the table, but I think that the context is almost clear from the question and the partial answer of it.