That person is using the hidden premise that everything which is true is also proven. This is incorrect, which can be pointed out in at least four ways:
If everything that is true is proven, research and universities (esp. in math and friends) would be useless. However, they are still used. Therefore, there are unproven truths.
Every proof in history shows that before that proof there was an unproven truth (the thing proven). So, to say that there are no unproven truths any longer at this point in time is making a very arrogant claim, which can't really be argued. Why would we be at such a special moment in time that exactly now there are no unproven truths?
If everything that is true is proven, that implies that everything that is true can be proven. However, there are some things for which no proof exists for either the positive or the negative claim (think about the God question, P=NP? and others).
There is a possibility that some things are neither true nor false, but can also be 'unknown' or even 'unknowable' among true, false and unknown. The argument assumes that these two other possibilities don't exist, without providing a reason for that.
Of course, this just considers the most general form of the argument. If you were discussing something which has been extensively studied and it can be reasonably assumed that everything which is true is proven in that field, his argument may actually be reasonable. However, in a strict sense it won't be correct.
By the way, an argument from silence usually considers (the absence of) historical documents, not of proofs. What you have is an argument from ignorance. The argument from silence does not always have to be a fallacy. For example, we might claim that since there is no mentioning of the exodus in Egyptian sources (while Egyptians kept very good records), the account in the book Exodus is probably historically incorrect. The argument from ignorance is always a fallacy.