Exists Subset which is not Element/Proof 1
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Theorem
Let $S$ be a set.
Then there exists at least one subset of $S$ which is not an element of $S$.
Proof
Let $S$ be a set.
Let $T$ be the set of all elements of $S$ which do not contain $S$ as elements.
Then by the corollary to Russell's paradox $T$ itself cannot be an element of $S$.
This $T$ is the required subset.
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 9$ Zermelo set theory