This is how definitions work. Definitions would get absurdly long and difficult to read if we defined everything in terms of first principles. I could concisely describe a solvable group as a group having a subnormal serious whose factor groups are all abelian. If I have to go back and explain group and subnormal series and factor groups and abelian it ballloons to a page in length, and those are all concepts that are useful elsewhere is well.
Presumably that author wasn't just defining things cyclically and had defined cardinality elsewhere. You'd just have to go back and look it up.