= Solution
An uncountable cardinal $\kappa$ is <measurable cardinal>[measurable] when it carries a <nonprincipal ultrafilter> $U$ that is <kappa-complete filter>[$\kappa$-complete]. For an inaccessible $\lambda>\kappa$, the cardinal $\kappa$ is <one-strong cardinal>[1-strong] when there is an <elementary embedding>
$$
j:V_\lambda\longrightarrow M
$$
into a transitive model, with <critical point of an elementary embedding>[critical point] $\kappa$ and $V_{\kappa+1}\subseteq M$.
The fundamental theorem on measurable cardinals constructs from $U$ the well-founded <ultrapower> and its <ultrapower embedding> $j:V_\lambda\to M$, whose critical point is $\kappa$. The embedding fixes $V_\kappa$. If $A\in V_{\kappa+1}$, then $A\subseteq V_\kappa$, and <elementary embedding>[elementarity] gives
$$
A=j(A)\cap V_\kappa.
$$
Both $j(A)$ and $V_\kappa$ belong to the transitive target, so $A\in M$. Thus $V_{\kappa+1}\subseteq M$, proving that every measurable cardinal is <measurable cardinal is one-strong>[1-strong].
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