Special well ordering on every set

Question

Can someone explain to me the last line? How is this the desired well ordering?

Answer 1

Let $\leq_0$ be the restriction of the initial order on $B$ to $B_{a_0}$. Then $(B_{a_0},\leq_0)$ has the desired property. Moreover, $\psi:B\to B_{a_0}$ is an order isomorphism between $(B,\leq^*)$ and $(B_{a_0},\leq_0)$, so $(B,\leq^*)$ will also have the desired property.

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EDIT: Let's see why $B_{a_0}$ has the desired property: if $x\in B_{a_0}=\left\{y:y<a_0\right\}$, then $y<a_0$. Since $a_0=\min\left\{a:|B_a|=|B|\right\}$, then $y\not\in\left\{a:|B_a|=|B|\right\}$, so $|B_y|<|B|=|B_{a_0}|$. This is precisely what we want.