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 Finitary alternating group (?) on $S$ is a Monolith (?) (a nontrivial normal subgroup contained in every nontrivial normal subgroup) in the 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. Alternating groups are simple: The alternating group on a finite set of size at least $5$ is simple.
2. Finitary alternating groups are simple: The finitary alternating group on an infinite set is simple.
3. Simple normal implies minimal normal
4. Finitary alternating group is centralizer-free in symmetric group
5. Self-centralizing and minimal normal implies monolith
6. 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 $G$.