Element of Every Transitive-Closed Class is Ordinal

Theorem

Let $x$ be a set such that $x$ is an element of every transitive-closed class.

Then $x$ is an ordinal.

Proof

This theorem requires a proof. In particular: Not even sure if it actually is true yet. The exercise as set in S&F asks the question whether it is or not. If it turns out not true, we rename this page. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 2$ Ordinals and transitivity: Exercise $2.1$