Personally born in the UK I can't fully comment on this issue but on my few (12 so far) trips to San Siro, I've bumped into fans of the big three clubs in Italy and sure, the Ultras are to be avoided but the usual everyday fan would never, ever support another of the big 3 in any competition, but then at the same time they'd not be ridiculous and call it utter hatred until it came to a match between the two. This is just my limited experience, I usually associate complete hatred with the younger fan of the club. The most interesting case I found was a Juve fan - about 35-40 years old - on the plane back from Milan after the Bologna game last season, he said he had a season ticket for 14 years for Juve and although he really disliked Milan, it was Inter he hated post-Calciopoli far more than Milan, then laughed at 'smaller' clubs like Roma who liked to whine about all the decisions all the time.
The way I call it is this - imagine a team, some random team you have no opinion about, like AS Monaco for example, absolutely humiliated the team you support, by a ridiculous goal margin, and then came up to face Milan or Inter, you'd still support Monaco right, and that is how I see it as a Milan fan, I'd still support them against Juventus or Inter, that's just the way it is and should be, however I don't call it hatred.
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