I'm sure there's a rule about this, but I couldn't remember nor find it. However, it's quite simple; "That book is mine" = "Den bog er min." There aren't really any words for 'mine', other than 'min/mit/mine'. You can re-phrase the sentence; "Den bog er min." = "Det er min bog." ("That is my book.") In most cases, I would say the latter sounds more pleasant and is the best option to use. So, re-arrange the words in the sentence; "That book is mine." = "Det er min bog." Btw - glad you caught the connection between 'en/et' and 'min/mit'.

Thanks for the comment.

Min is used for uter words in singular (en bog) Mit for neutrer words in singular (et hus) Mine is used for plural (mine børn)