TITLE: Examples of ZFC theorems proved via forcing 
QUESTION [16 upvotes]: This is an old suggestion of Joel David Hamkins at the end of his answer to this question: Forcing as a tool to prove theorems
I just noticed it while trying to understand his answer. But indeed it would be nice to have a big list of $ZFC$ theorems that were proven first by forcing. 
A very well known example is Silver's Theorem about the fact that the $GCH$ can't fail first at a singular cardinal of uncountable cofinality (say for instance $\aleph_{\omega_1}$), I had read somewhere (Jech, maybe) that Silver proved it first using forcing. 
Also if anyone knows theorems of pcf theory that were first proven using forcing, please post them.

REPLY [6 votes]: Another example around a theorem of Gitik and Shelah: If there is a total extension $m:\mathcal{P}([0, 1]) \rightarrow [0, 1]$ of Lebesgue measure then there is an uncountable set of reals all of whose Lebesgue null subsets are countable.
Proof sketch: Force with the measure algebra of $m$. By (a weak form of) Gitik-Shelah theorem (whose original proof uses generic ultrapower arguments too), this forcing adds at least $\omega_1$-many random reals in the generic extension. But this set of randoms is also in the generic ultrapower so by elementarity we have such a set in $V$.
For some open questions along these lines see Fremlin's survey article on real valued measurable cardinals and also his problem list.