Fix $E \in \mathcal{F}$. Now result R7: Empty set is subset of every set and the definition of a D84: Sigma-algebra guarantee the inclusions $\emptyset \subseteq E \subseteq X$. Thus R975: Isotonicity of unsigned basic measure and again the definition of a measure imply \begin{equation} 0 = \mu(\emptyset) \leq \mu(E) \leq \mu(X) \end{equation} Since $E \in \mathcal{F}$ was arbitrary, this implies that $\mu(F) \in [0, \mu(X)]$ for all $F \in \mathcal{F}$, which finishes the proof. $\square$