I'm teaching myself set theory, and I'm not sure how detailed I should be when asked to prove things.
Here is my proof that $A\subseteq A$ (the subset relation is reflexive):
$A \subseteq B$ iff $((x \in A) \implies (x \in B))$
$A \subseteq A$ iff $((x \in A) \implies (x \in A))$
$(x \in A) \implies (x \in A)$ is always true, as something implying itself must be true (is there a formal way to write this?)
Hence $ A \subseteq A$ is always true.
Is this proof formal enough, and does it contain the right amount of detail?