# Which axioms of ZF are needed for Cantor's theorem?

## Main Question or Discussion Point

Self explanatory. The Cantor's theorem which am referring to is that the cardinality of the power set of any set is greater than that of the set.

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CRGreathouse
Homework Helper
The Metamath proof of Cantor's theorem (canth) uses basic logic*, plus the following:
• Axiom of Extensionality (ax-ext)
• Axiom of Separation (ax-sep)
• Null Set Axiom (ax-nul)
• Axiom of Power Sets (ax-pow)
• Axiom of Pairing (ax-pr)
• Axiom of Union (ax-un)
The Axioms of Replacement, Regularity, and Infinity of ZF are not used in the proof.

* Propositional calculus (ax-1, ax-2, ax-3, ax-mp), basic predicate calculus (ax-4, ax-5, ax-6, ax-7, ax-gen), the equality and substitution rules (ax-8, ax-9, ax-10, ax-11, ax-12, ax-13, ax-14), and the second Axiom of Quantifier Introduction (ax-17).
** The page also lists the Axiom of Distinct Variables (ax-16) and the old Axiom of Variable Substitution (ax-11o), but these have been proven redundant with the others.

Fascinating. Thanks.