Suppose $\mathbb{P}$ is a notion of forcing in the ground model $V$, and $X$ is a set which is in $V[G]$ for every $\mathbb{P}$-generic filter $G$. Then $X\in V$ already, by a fairly simple (if tedious) argument.


I'm working on a paper in which this fact is mentioned, and I would like to cite it properly; however, I can't seem to find any citation for it, and I don't recall when I learned the argument. So my question is:



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> What is a citation for "if $X$ is in every forcing extension by some poset $\mathbb{P}$, then $X$ is already in the ground model?"
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Who proved "sets in every generic are already in the ground model?"
The result is due to Solovay, see 


"A model of set theory in which every set of reals is Lebesgue measurable"


In Lemma 2.5 of the above paper, Solovay shows that if $G$ and $H$ are mutually generic filters for a forcing notion $P,$ then $V[G]\cap V[H]=V.$ From this result your statement follows immediately.


Note that Solovay's result is even more general than what I said above, and it says if $G\times H$ is $P\times Q-$generic over $V$ (where $P, Q \in V$), then $V[G]\cap V[H]=V.$


