Cardinal of Cardinal Equal to Cardinal/Corollary

From ProofWiki
Jump to navigation Jump to search

Theorem

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

Let $x$ be an ordinal.


Then:

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


Proof

Necessary Condition

Suppose $x = \card x$.

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

By definition of cardinal class:

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

It follows that $x \in \NN$.

$\Box$


Sufficient Condition

\(\ds x\) \(\in\) \(\ds \NN\) by hypothesis
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds \card y\) Definition of Class of 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

$\blacksquare$


Sources