If you want to go to the world's best school for mine engineering, Yale shouldn't be on your list to begin with. It's a sign-off for pedigreed elites, not a trade school.
Stanford also confers more degrees in social sciences and "interdisciplinary studies" (i.e. degrees in having graduated from Stanford) than in engineering and computer science, and has done for some years now. I don't doubt you can obtain an excellent education in those fields at Stanford, but it's been a long time since that was what Stanford was for.