The first part of your argument is false as well.:

A transgender woman lives as a woman today, but was thought to be male when she was born. A transgender man lives as a man today, but was thought to be female when he was born. Some transgender people identify as neither male nor female, or as a combination of male and female. There are a variety of terms that people who aren't entirely male or entirely female use to describe their gender identity, like non-binary or genderqueer.