I believe you're confusing arrogance and closedmindedness.

Well, maybe that's the question. They're different, and you can have one without the other, but do they cooccur above chance? Maybe arrogance reduces your exposure to the occasional clever ideas that will inevitably come from people you've dismissed. That isn't closemindedness, it something more like as-if-closemindedness, but it would come to the same thing.