Definition df-aleph 9946

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 10072, alephsuc 10074, and alephlim 10073. 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 9942	. 2 class ℵ
2		char 9562	. . 3 class har
3		com 7855	. . 3 class ω
4	2, 3	crdg 8417	. 2 class rec(har, ω)
5	1, 4	wceq 1539	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10071  aleph0  10072  alephlim  10073  alephsuc  10074