Definition df-aleph 9856

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 9980, alephsuc 9982, and alephlim 9981. 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 9852	. 2 class ℵ
2		char 9465	. . 3 class har
3		com 7810	. . 3 class ω
4	2, 3	crdg 8342	. 2 class rec(har, ω)
5	1, 4	wceq 1542	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9979  aleph0  9980  alephlim  9981  alephsuc  9982