Cardinal Number Equivalence or Equal to Universe
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Theorem
Let $S$ be a set.
Let $\card S$ denote the cardinal number of $S$.
Let $\mathbb U$ denote the universal class.
Then:
- $S \sim \card S \lor \card S = \mathbb U$
Proof
By Condition for Set Equivalent to Cardinal Number:
If $\exists x \in \On: S \sim x$, then:
- $S \sim \card S$
If $\neg \exists x \in \On: S \sim x$, then:
\(\ds \bigcap \set {x \in \On : S \sim x}\) | \(=\) | \(\ds \bigcap \O\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbb U\) | Intersection of Empty Set |
And thus by the definition of cardinal number:
- $\card S = \mathbb U$
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.10$