Understanding a proof involving Zorn's lemma.

Question

Am trying to understand this proof. I get everything except why must $X_1$ and $X_2$ necessarily be uncountable?

Thanks a lot for your help.

Answer 1

Let $Z=X_1\cup X_2$. If $X\setminus Z$ has more than one element, we can form an ordered pair with two elements of $X\setminus Z$ and add it to $A^*$ to get to an element of $P$ strictly larger than $A^*$, contradicting the maximality of $A^*$. Thus, $|X\setminus Z|\le 1$, and $Z$ must therefore be uncountable.

By construction there is a bijection between $X_1$ and $X_2$, so $|X_1|=|X_2|$. $Z=X_1\cup X_2$, so if $X_1$ and $X_2$ were countable, $Z$ and therefore $X$ would be countable. Thus, $X_1$ and $X_2$ must be uncountable.