Cardinal of Cardinal Equal to Cardinal/Corollary

From ProofWiki
Jump to navigation Jump to search


Let $\NN$ denote the class of all cardinal numbers.

Let $x$ be an ordinal.


$x \in \NN \iff x = \card x$


Necessary Condition

Suppose $x = \card x$.

Then $x = \card y$ for some $y$ by Existential Generalisation.

By definition of class of all cardinals:

$\NN = \set {x \in \On: \exists y: x = \card y}$

It follows that $x \in \NN$.


Sufficient Condition

\(\ds x\) \(\in\) \(\ds \NN\) by hypothesis
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds \card y\) Definition of Class of All Cardinals
\(\ds \leadsto \ \ \) \(\ds \card x\) \(=\) \(\ds \card y\) Cardinal of Cardinal Equal to Cardinal
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds \card x\) Equality is Transitive