Axiom of Choice implies Zorn's Lemma/Proof 3
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Theorem
Let the Axiom of Choice be accepted.
Then Zorn's Lemma holds.
Proof
We have that Axiom of Choice implies Tukey's Lemma.
Then we have Tukey's Lemma implies Zorn's Lemma.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles: Proposition $5.8$