Definition df-aleph 9629

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 9753, alephsuc 9755, and alephlim 9754. 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.)

Assertion

Ref	Expression
df-aleph	⊢ ℵ = rec(har, ω)

Detailed syntax breakdown of Definition df-aleph

Step	Hyp	Ref	Expression
1		cale 9625	. 2 class ℵ
2		char 9245	. . 3 class har
3		com 7687	. . 3 class ω
4	2, 3	crdg 8211	. 2 class rec(har, ω)
5	1, 4	wceq 1539	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9752  aleph0  9753  alephlim  9754  alephsuc  9755