Here the following is stated:

It's a basic fact in $p$-adic Hodge theory that any 2-dim. absolutely irreducible $G_{\mathbb Q_p}$-representation with distinct Hodge-Tate weights is uniquely determined by $a_p$.

Could somebody explain in maximum detail why is this true? Taking inspiration from the famous ELI5 communication style, explain like I am a generic first-year grad student.