Show that same cardinal number defines and equivalence relation

Question

Same cardinal number must satisfy all three properties : symmetry, reflexivity, transitivity

Symmetry

Suppose bijection $f:A→A$, Then by definition, |A| = |A|

Reflexivity

Suppose bijections $f:A→B$ and $f:B→A$, Then |A| = |B| and |B| = |A|

Transitivity

Suppose bijections $f:A→B$ and $f:B→C$ Then |A|= |B| and |B| = |C|

I think I did this right but I feel as though this is too simple of a proof or I have the definitions wrong