I have an exercise in my course, which asks to prove $A \cup B = B \iff A \subseteq B$.
My proof is: Let $A \nsubseteq B$, that is, $\exists a \in A : a \notin B$. Then from the definition follows $a \in A \cup B = B$, in contradiction to the initial assertion. $\square$
Usually I see that it's much more rigorous to prove $\implies$, then $\impliedby$, but I'm not sure, if that's only an option or a strict rule — and specifically if my proof does the job in both directions or there are some gaps that I don't recognize. My script suggests a really long 10+ lines proof using the 'both directions style', but I myself don't really see this necessity at least here.
This being said, is it always a must to prove the 'iff' in both directions?