auxiliary of prediction, "be obliged, be necessarily impelled," from Old English moste, past tense of motan "have to, be able to," from Proto-Germanic *motanan (source also of Old Saxon motan "to be obliged to, have to," Old Frisian mota, Middle Low German moten, Dutch moeten, German müssen "to be obliged to," Gothic gamotan "to have room to, to be able to"), perhaps from PIE root *med- "take appropriate measures," but this old suggestion lately has been doubted. Used as present tense from c. 1300, eventually displacing motan, from the custom of using past subjunctive as a moderate or polite form of the present.
<a href="https://www.etymonline.com/word/mustn't">Etymology of mustn't by etymonline</a>
Harper, D. (n.d.). Etymology of mustn't. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/mustn't