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I'm looking for nice and straightforward applications of the cantor theorem which says that for each set $A$, we have $|A|\lt |P(A)|$. I know this example: $|P(\mathbb N)|=\mathbb R$, but I find the proof tedious using binary system, maybe if someone could find another better proof, it may be really helpful as well.

Thanks in advance

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Corollary: Given a cardinality, there is one strictly larger.

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One non-trivial application is that there is no set $S$ of all sets, because if there were then the power set of $S$ would have to equal $S$, contradicting that $|P(S)| > |S|$. This fact that there is no set of all sets can be proved other ways but the Cantor theorem gives a rather simple proof.

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