Reminds me of this article here: http://tirania.org/blog/archive/2011/Feb-17.html
She should not feel uncomfortable just because she is wrong on some minor fact. If both people accept the idea that they are often wrong it will be much easier for them to fix problems with the relationship later on.
Say she is making a point about cheese, and says something about the population of France. She might be wrong, but who cares? Or you can shout her down and criticize here, making her uncomfortable. Showing how smart you are, and how much of an asshole you are as well.