I'm studying Enderton's Elements of Set Theory and in the page 129 he defines what it means two sets being equinumerous:
After that in the page 136 he defines cardinality:
Why doesn't he define cardinality directly saying that $card (A)=card(B)$ iff there is a one-to-one function from $A$ onto $B$?
Thanks
That is not a definition of cardinality, and Enderton does not say it is. What he actually says is that we want to define $\operatorname{card} A$ in such a way as to make $\operatorname{card} A = \operatorname{card} B$ iff $A \approx B$. He goes on to say that there is not a simple way to do this, but (in effect) that he will proceed as if such a definition had been made, while promising that a proper definition will eventually be given. The actual definition of $\operatorname{card} A$ occurs on page 197.
As to Enderton's specific exposition, the notation $A \approx B$ is useful on its own. It's more compact, and you don't have any awkward definitional issues to rigorously justify using it. And once Enderton defined it, it made sense to reuse it to explain the $\operatorname{card}$ notation. This is similar to the general practice of using an already defined concept to define a related concept.