Here is what i think.
Prince:
Argosax was the King of Hell.
Mundus was his son and he quickly grew more powerful and then overthrowed Argosax to become King.
Not Prince:
Argosax was the King of Hell.
Mundus was born and he rose to power quickly eventually overthrowing Argosax.
Don't know if that helps.
That actually does kinda help. Both theories are plausible. It would also explain why Argosax was so weak compared to Mundus. Mundus ruled because he was obviously the superior demon.