$H, K$ are subgroup of $G$. If $H \cup K \leq G$, then $H \subseteq K$ or $K \subseteq H$? Is the statement True or False? 

Let $H,K$ be subgroups of a group $G$.
  If $H \cup K \leq G$, then $H \subseteq K$ or $K \subseteq H$.

Need help with this question.
 A: Assume neither is a subset of the other. Then there exist $h\in H\setminus K$ and $k\in K\setminus H$. Is the element $hk$ in either subgroup?
A: To determine if this statement is either true or false, we would have to find one instance where the given statement is not true. In this implies statement, since we know that the union of subgroups of G can at most be equal to G, then we know the first part of this implies statement is true since $H$ and $K$ are subgroups of $G$. Then, for the second part of the implies statement, if we assume that $H$ and $K$ are disjoint subgroups of $G$, we then can show that $H$ and $K$ cannot be subsets of one another since they do not share any elements. Hence, the second part of this statement would be false. A $T$ $->$ $F$ statement is always false and therefore that would make this statement false.
Something to consider: Had we switched the clauses of the statement to: If $H \subseteq K$ or $K \subseteq H$, then $H \cup K \leq G$ then the statement would be vacuously true.
