How do I prove cardinality is well-defined?

Question

I define equinumerous and cardinality in this way:

$A$ and $B$ are equinumerous (written $A\sim B$) if there is a bijection between them.

We say $card(X)=card(Y)$ if $X\sim Y$.

I would like to prove the cardinality is well-defined. This is how I did (I'm using the symmetry and transitive properties):

$X\sim Y$ and $X\sim Z\implies Y\sim Z\implies card(Y)=card(Z)$.

Am I correct? Do I need to check if the concept of equinumerous is well-defined as well?

Thanks

Answer 1

You could also think that when you say "$card(X)=card(Y)$ iff $X\sim Y$", you're meaning:

"Assign every set $X$ a number called $card(X)$ in such a way that $card(X)=card(Y)$ iff $X\sim Y$"

In fact, you need to prove that $\sim$ is an equivalence relation in order to have a good definition of this number. Later on, you define $card(X)$ for known sets as $\{1,...,n\}$ or $\mathcal{P}(\mathbb{N})$, for example.