Double Nope.
Prince wrote it, but originally for a funk band named The Family.
Prince wrote it and the Family recorded it in 1985. Prince's live version didn't come until 1993.
Sinead O'Conner recorded it in 1990.
After Prince passed, a "very first" recording was released, done by Prince, in 1984 at the Flying Cloud Drive Warehouse in Eden Prairie, Minnesota.
O'Conner's version said to be the best but let's face it, Cornell nailed it too.
Here's the OG from The Family...