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**Ester** 1) Let's choose arbitrary $\displaystyle \beta \in \alpha$ and assume that $\displaystyle \gamma \in \beta$. $\displaystyle \beta \preceq A$, so there exists an injection $\displaystyle f:\beta \rightarrow A$. $\displaystyle \gamma \in \beta $ so $\displaystyle \gamma < \beta$. There for function $\displaystyle g:\gamma \rightarrow A$ is also injection. That means that $\displaystyle \gamma \preceq A$, so $\displaystyle \gamma \in \alpha$. $\displaystyle \alpha$ is a set of ordinals and it is transitive, so $\displaystyle \alpha$ is ordinal.