If you’ve ever been to a Penn-Cornell game, you’d know that the chant “STATE SCHOOL. STATE SCHOOL.” is used by Quaker fans to mock Cornell.
One of the most significant aspects of the Ivy League is that it was the first (or at least most prominent) Division I conference to ban athletic scholarships.
Perhaps you meant Baylor, the only private university in the Big 12. Or Vanderbilt, the only private university in the SEC. Or Northwestern, the only private university in the Big 10.