Definition df-aleph 9955

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 10081, alephsuc 10083, and alephlim 10082. 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 9951	. 2 class ℵ
2		char 9571	. . 3 class har
3		com 7864	. . 3 class ω
4	2, 3	crdg 8423	. 2 class rec(har, ω)
5	1, 4	wceq 1534	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10080  aleph0  10081  alephlim  10082  alephsuc  10083