Normality-large and minimal normal implies monolith

Statement
A fact about::minimal normal subgroup of a group that is normality-large, must be a fact about::monolith: it is contained in every nontrivial normal subgroup.

Facts used

 * 1) uses::Normality is strongly intersection-closed

Proof
Given: A group $$G$$, a minimal normal subgroup $$N$$ such that if $$N \cap M$$ is trivial for any normal subgroup $$M$$, then $$M$$ is trivial.

To prove: For any normal subgroup $$M$$, either $$M$$ is trivial or $$N \le M$$.

Proof: Since both $$N$$ and $$M$$ are normal, fact (1) tells us that $$N \cap M$$ is normal. So $$N \cap M$$ is a normal subgroup of $$G$$ contained in a minimal normal subgroup $$N$$. So there are two cases:


 * $$N \cap M = N$$: In this case, $$N \le M$$.
 * $$N \cap M$$ is trivial: In this case, the normality-largeness of $$N$$ tells us that $$M$$ is trivial.

This completes the proof.