Successor of Ordinal Smaller than Limit Ordinal is also Smaller/Proof 2

Proof
Because $\lambda$ is a limit ordinal:
 * $\lambda \ne \alpha^+$

Moreover, by Successor of Element of Ordinal is Subset:
 * $\alpha \in \lambda \implies \alpha^+ \subseteq \lambda$

Therefore by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal:
 * $\alpha^+ \subset \lambda$ and $\alpha^+ \in \lambda$