Axiom of Choice implies Maximal Principles

Theorem

Let the Axiom of Choice be accepted.

Then the Maximal Principles hold.

Proof

From Maximal Principles are Equivalent, it is sufficient to demonstrate that any one of them is implied by the Axiom of Choice.

Indeed, we have several such theorems:

• Axiom of Choice implies Kuratowski's Lemma
• Axiom of Choice implies Tukey's Lemma
• Axiom of Choice implies Zorn's Lemma
• Axiom of Choice implies Hausdorff's Maximal Principle

$\blacksquare$

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