Unordered Pairs Exist
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The validity of the material on this page is questionable. In particular: This seems to suggest that a set containing classes is an element of the universal class, which cannot happen in the context of NBG. If this is a genuine theorem, we need to establish what its context actually is. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. 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 {{Questionable}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Theorem
Let $A$ and $B$ be classes.
Then:
- $\forall A, B: \set {A, B} \in U$
where $U$ is the universal class.
Proof
\(\ds \forall A, B: \exists x: \forall y: \, \) | \(\ds \leftparen {y \in x}\) | \(\iff\) | \(\ds \rightparen {y = A \lor y = B}\) | Axiom of Pairing | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall A, B: \exists x: \, \) | \(\ds x\) | \(=\) | \(\ds \set {y: y = A \lor y = B}\) | Definition of Set Equality | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall A, B: \, \) | \(\ds \set {y: y = A \lor y = B}\) | \(\in\) | \(\ds U\) | Element of Universe | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall A, B: \, \) | \(\ds \set {A, B}\) | \(\in\) | \(\ds U\) | Definition of Doubleton |
$\blacksquare$
Also see
Sources
- 1963: Willard Van Orman Quine: Set Theory and Its Logic: $\S 7.10$