I don’t want my ex’s girlfriend around our child?

Okay so before I get called a bitter babymom or the thousandth time, let me explain... I was 6 months pregnant & my ex was cheating on me, he wanted me to move in with him & made plans for us to be living together all while he was with his ex girlfriend of 3 years behind my back. Going through his Facebook & seeing he was with her when he wasn’t with me was not the only thing I’ve discovered through his messages. She said stuff like “I hate her” “I hate you so much for getting her pregnant twice” “that was suppose to be my baby” “it’s going to suck that you have to pay child support, it’s going to fuck things up for us when we live together” “she’s a small minded girl” “she’s weak minded” “come inside me & we will both get scared” “I don’t want to be with you because I can’t have kids with you because you already have a kid” “how are you going to treat my babies with her baby” THEY HAVE NO KIDS TOGETHER, but she was constantly comparing my living breathing child to non existent babies, bitched about him having to pay for child support, wanted to get pregnant just because I was pregnant & when I looked even further into their messages she said she didn’t want kids before I even got pregnant. So all of a sudden she wants babies just because I was pregnant! It’s so obvious she was jealous of my child. Now throughout my entire pregnancy he cheated on me with her & they got back together before I even gave birth, now she posts about being a step mom & messages me saying she’s my babies mom & can't wait to be apart of my sons life. He had my baby for a visit while she was with him after I made it clear I don’t want her around my son. This is reasonable right? I thought I was going to have a home & family now she expects to play happy family with my child after everything she said & done. I believe she only does this because she wants to make me mad otherwise she wouldn’t throw it in my face that she’s going to be my kids step mom forever.
Updates:
3 mo
Forgot to add as soon as I found out about the cheating he left me & they now live together.
I don’t want my ex’s girlfriend around our child?
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