Let $A \subseteq P(\omega)$, where $\omega$ is the set of all natural numbers and $P(\omega)$ is the power set of $\omega$. If $\langle A,\subseteq\rangle$ is a well ordered set how can you prove that $A$ is a countable set.
Let $A$ be a set which elements are closed sets of real numbers. If $\langle A,\subseteq\rangle$ is a well ordered set how can you prove that $A$ is a countable set.
How can you prove that Zorn's lemma (so and the axiom of choice) is equivalent to that for every partially ordered set $\langle A,\le\rangle$ that satisfies Zorn's condition, for every $b\in A$ there exists a maximum element $a$ for which $b\le a$.
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