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?