Well-foundedness proof for $\Pi^{1}_{1}$-reflection

In the lecture notes it is shown that an ordinal $\psi_{\Omega}(\varepsilon_{\mathbb{S}^{+}+1})$ is an upper bound for the proof-theoretic ordinal of a set theory ${\sf KP}\omega+(M\prec_{\Sigma_{1}}V)$. In this note we show that ${\sf KP}\omega+(M\prec_{\Sigma_{1}}V)$ proves the well-foundedness up to $\psi_{\Omega}(\omega_{n}(\mathbb{S}^{+}+1))$ for each $n$.