Definition df-aleph

Description: Define the aleph function. Our definition expresses Definition 12 of Suppes p. 229 in a closed form, from which we derive the recursive definition as theorems aleph0 , alephsuc , and alephlim . The aleph function provides a one-to-one, onto mapping from the ordinal numbers to the infinite cardinal numbers. Roughly, any aleph is the smallest infinite cardinal number whose size is strictly greater than any aleph before it. (Contributed by NM, 21-Oct-2003)

	Ref	Expression
Assertion	df-aleph	$$⊢\aleph =\mathrm{rec}\left(\mathrm{har},\mathrm{\omega }\right)$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cale	$$class\aleph$$
1		char	$$class\mathrm{har}$$
2		com	$$class\mathrm{\omega }$$
3	1 2	crdg	$$class\mathrm{rec}\left(\mathrm{har},\mathrm{\omega }\right)$$
4	0 3	wceq	$$wff\aleph =\mathrm{rec}\left(\mathrm{har},\mathrm{\omega }\right)$$