Definition df-aleph 9857

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 9981, alephsuc 9983, and alephlim 9982. 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 9853	. 2 class ℵ
2		char 9466	. . 3 class har
3		com 7811	. . 3 class ω
4	2, 3	crdg 8343	. 2 class rec(har, ω)
5	1, 4	wceq 1542	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9980  aleph0  9981  alephlim  9982  alephsuc  9983