Since $f$ is nonzero then, by construction, one has the pointwise inequality
\begin{equation}
\lambda I_{\{ f \geq \lambda \}} \leq f
\end{equation}
Result
R1214: Isotonicity of unsigned basic integral allows us to integrate each side and preserve the inequality, while result
R1213: Linearity of unsigned basic integral allows us to factor the constant $\lambda > 0$ out of the integral. We obtain
\begin{equation}
\lambda \int_X I_{\{ f \geq \lambda \}} \, d \mu \leq \int_X f \, d \mu
\end{equation}
Applying now
R1242: Unsigned basic integral is compatible with measure on the left hand side, we conclude
\begin{equation}
\lambda \mu( f \geq \lambda) = \lambda \int_X I_{\{ f \geq \lambda \}} \, d \mu \leq \int_X f \, d \mu
\end{equation}
The result then follows by multiplying each side by the constant $1 / \lambda$. $\square$