This is a great question. In many ways, when people are proud, they feel that their way of life, being, culture, or anything else is better than someone else's. The key point is that pride is always competitive. It has to, by definition, have an opponent and seek to win. C. S. Lewis made this point long ago:
Pride gets no pleasure out of having something, only out of having more of it than the next man... It is the comparison that makes you proud: the pleasure of being above the rest. Once the element of competition is gone, pride is gone.
This is why pride is one of the worst vices. For example, it is the most dangerous sin on the list of the seven deadly sins. Also the Romans called it Superbia and Greeks Hubris.
Now based on these points, it is easy to see how pride can lead to prejudice. If you are a proud person, then you cannot really consider others, their points of view, and love them.
I would say that with pride a person does not want to admit fault or is sometimes to far in their ego. The way prejudice can fall into this is a person may have a.perception that is incorrect or misguided and become to judgemental or feel to.strongly with their perception.