S.S.
Oh dear. At this point I would give up, totally, on the idea of impressing idealistic egalitarian views on the little guy. He's a happy self-centered little primate full of the joy of accomplishment... BUT you can address it as a matter of good manners.
--Teach him how to shake hands and say, "good game" when he wins--and loses. Play games and run races with him so he gets used to the idea of winning, and losing.
--If he is sad about losing something, ask him to think of what he learned from it.
--Praise trying, not just success.
--When he is bragging about something being smarter or faster or stronger or bigger, or his toys or his house or money or anything), encourage empathy. "How would you feel if a big kid came over to you and laughed at you for being little?" "How would you feel if someone said she was better than you because her house was bigger"? "Would you want to play with a little boy if he was always talking about how he was stronger?"
--If he says something belittling to another person in front of you, model empathy for him by example. "I am so sorry that my son said something unkind about your dog... I apologize. I feel very bad for you. Your dog is adorable and looks very healthy. It doesn't matter who has the bigger dog." But don't force him to apologize.
--Brace yourself for the possibility that you will have a little guy whose feelings about winning and losing are rather different from yours : ) Sometimes it doesn't come from school.
--I think that avoiding competition might make it worse, because he will never get used to winning and losing gracefully. It will always be a "big deal."