Axiom of Choice implies Tukey's Lemma
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Theorem
Let the Axiom of Choice be accepted.
Then Tukey's Lemma holds.
Proof
Let us assume the truth of the Axiom of Choice.
From Axiom of Choice implies Kuratowski's Lemma, it follows that Kuratowski's Lemma holds.
From Kuratowski's Lemma implies Tukey's Lemma, it follows that Tukey's Lemma holds.
Hence the result.
$\blacksquare$
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 $5.6$