Does my attempt look fine or contain gaps? Thank you so much!
Theorem: If $\alpha$ and $\beta$ are ordinals. Then $\gamma=\alpha\cap\beta$ is an ordinal.
Proof:
- $\gamma$ is well-ordered
Since $\gamma\subseteq\alpha$ and $\alpha$ is well-ordered, $\gamma$ is well-ordered.
- $\gamma$ is transitive
If $\delta\in\theta\in\gamma$, then $\theta\in \alpha$ and $\theta\in \beta$. Then $\delta\in\theta\in \alpha$ and $\delta\in\theta\in \beta$. Then $\delta\in \alpha$ and $\delta\in \beta$ by the fact that $\alpha$ and $\beta$ are transitive. Hence $\delta\in\alpha\cap\beta=\gamma$.