Finitary alternating group is intermediately monolith in symmetric group

Statement
Let $$S$$ be an infinite set, or a finite set of cardinality at least $$5$$. Then, the fact about::finitary alternating group on $$S$$ is a fact about::monolith (a nontrivial normal subgroup contained in every nontrivial normal subgroup) in the fact about::symmetric group on $$S$$.

In fact, more is true: it is a monolith in every subgroup of the symmetric group containing it.

Facts used

 * 1) uses::Alternating groups are simple: The alternating group on a finite set of size at least $$5$$ is simple.
 * 2) uses::Finitary alternating groups are simple: The finitary alternating group on an infinite set is simple.
 * 3) uses::Simple normal implies minimal normal
 * 4) uses::Finitary alternating group is centralizer-free in symmetric group
 * 5) uses::Self-centralizing and minimal normal implies monolith
 * 6) uses::Normality satisfies intermediate subgroup condition

Proof
Given: A set $$S$$ of cardinality at least $$5$$. $$H$$ is the finitary alternating group on $$S$$. $$G$$ is any subgroup of the symmetric group on $$S$$, containing $$H$$.

To prove: $$H$$ is contained in every nontrivial normal subgroup of $$G$$.

Proof: Note that since $$H$$ is normal in the whole symmetric group on $$S$$, $$H$$ is also normal in $$G$$ by fact (6). Further, by facts (1) and (2), we see that $$H$$ is a simple group in all cases. Thus, $$H$$ is a simple normal subgroup of $$G$$. By fact (3), $$H$$ is a minimal normal subgroup of $$G$$. By fact (4), we conclude that $$C_G(H)$$ is trivial, so $$H$$ is, in particular, a self-centralizing subgroup of $$G$$. Fact (5) then tells us that $$H$$ is a monolith in $$G$$.