How to show in a proof that a statement is vacuously true because "if $\alpha$ then $\beta$", and also prove $\alpha$ is false, in a formal way? and also particularly, how to structure such proofs?

---

For example to vacuously prove $\forall A. \varnothing \subseteq A$.  

Speaking in natural language (I'm new to mathematics), what I want to formally say is:  

$\forall x.$ if $x \in \varnothing $ then $x \in A$.  

$x \notin \varnothing $: etc. etc.  

"if $x \in \varnothing $" is true because the condition is false, because $x \notin \varnothing $, therefore $x \in A$, vacuously.