OK I understand better your argument. However I still disagree.I think you may have misunderstood the private language argument. The argument uses memory skepticism to show that a private language is unintelligible even to the person for whom it is supposedly private.
If you mean that language (and thus probably the whole world) would be unintelligible within solipsism, that is probably true. But that does not entail that solipsism is false.
I don't think we can prove that language is intelligible. This is merely an unquestionned assumption on the basis of which we use language, just as the assumption that other people genuinely exist is an unquestioned assumption on the basis of which we communicate and live.