Why did Snape name himself "Half-Blood Prince" in Harry Potter 6?

Half-blood prince, Severus 6? 1 Snape. English is Half-blood Prince, which also means "half-prince blood".

Here is a pun in English, which stands for prince and prince.

He is because his mother's name is Prince Irene, which means prince in Chinese.

He is called the half-blood prince because his father is a muggle and his mother is a wizard.

He is a half-blood, so he is called the half-blood prince.