Theorem cardnum

Description: Two ways to express the class of all cardinal numbers, which consists of the finite ordinals in _om plus the transfinite alephs. (Contributed by NM, 10-Sep-2004)

	Ref	Expression
Assertion	cardnum	$$⊢\left\{x|\mathrm{card}\left(x\right)=x\right\}=\mathrm{\omega }\cup \mathrm{ran}\aleph$$

Proof

Step	Hyp	Ref	Expression
1		iscard3	$$⊢\mathrm{card}\left(x\right)=x\leftrightarrow x\in \left(\mathrm{\omega }\cup \mathrm{ran}\aleph \right)$$
2	1	bicomi	$$⊢x\in \left(\mathrm{\omega }\cup \mathrm{ran}\aleph \right)\leftrightarrow \mathrm{card}\left(x\right)=x$$
3	2	eqabi	$$⊢\mathrm{\omega }\cup \mathrm{ran}\aleph =\left\{x|\mathrm{card}\left(x\right)=x\right\}$$
4	3	eqcomi	$$⊢\left\{x|\mathrm{card}\left(x\right)=x\right\}=\mathrm{\omega }\cup \mathrm{ran}\aleph$$