My dad is a Michigan Phd and Harvard post doc. His response to Michigan men who say "we're a public Ivy, the Harvard of public universities" is to say "No Harvard grad has ever claimed to be from the Michigan of private schools"
Also, for what it is worth, Stanford is truly the Harvard of the west. Leland Stanford traveled to Cambridge fully determined to donate his entire fortune to Harvard. Upon arrival, he lobbied for a meeting with the president. After finally getting the meeting, Stanford, a modestly dressed man from the wilds of California was instantly shooed out of the office, the president scoffing that Harvard didn't need a small donation. Harvard's president had no idea the plainly dressed man was among the wealthiest men in California. Stanford got back on a train, headed west and founded the university that bares his name on a 3,000 acre parcel of his farm.