Abelian and MWNSCDIN implies SCDIN

Statement
If $$H$$ is an abelian MWNSCDIN-subgroup of a group $$G$$, then $$H$$ is a SCDIN-subgroup of $$G$$.

Proof idea
The idea is that since $$H$$ is abelian, any subset of size one in $$H$$ is a normal subset. Thus, given any subset $$A$$ of $$H$$, we can view $$A$$ as the collection of its singleton subsets, and use the fact that $$H$$ is MWNSCDIN to conclude that it is SCDIN.