I'm going to make my mom give up our family cats
That feels very unnecessary. Why would you give your cats away?
Mom gets too weak and emotional as she can't take care of them anymore and I hate that
You shouldn't force your mother to give up her pets for no good reason. In fact, the cats may be the reason she's not even more weak and emotional. Her cats may be like family to her and taking them away will make her worse.
I don't care about her and I'll show her.
What you can do is to try and maybe help her around the house if you have time and take care of the cats. If you can't do so, maybe try to find a caretaker of the cats or at least someone to maintain and feed them everyday. I do hope you find a resolution for this and limit the blame with your mother.