$(∀x < y)(x ∈ β) ⇒ y ∈ β$, Prove$\beta=\alpha$

Question

Here $α$ is an ordinal number and $β ⊆ α$ has the property $(∀x < y)(x ∈ β) ⇒ y ∈ β$ for every $y ∈ α$. Prove that $β = α$.

Any help will be appreciate. Thanks in advance!

Answer 1

Let $\lambda$ be the least element in $\alpha$ such that $\lambda\not\in\beta$. Then $\forall \delta <\lambda,\delta\in\beta$ and so by the property, $\lambda\in\beta\bot$