Correct me if I am wrong, but -
Slavic people called themselves "slavs" deriving it from that very "slav" = glory. Around the XIV century, however, Slavs living near the Black Sea were often taken as slaves by vikings. Eventually, the notions Slav (ethnic) and "a person owned by somebody else" became equivalent, so the latter was "named" slave. Sad
as it is (because I am Slavic), the word "slave" comes from "slav", not vice versa.
Punto final: I always thought that Bratislava (the Slovk capital) is derived from "brat" - brother and "slav". Wouldn't swear upon it, though