Definition df-aleph 9853

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 9977, alephsuc 9979, and alephlim 9978. 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 9849	. 2 class ℵ
2		char 9462	. . 3 class har
3		com 7808	. . 3 class ω
4	2, 3	crdg 8339	. 2 class rec(har, ω)
5	1, 4	wceq 1542	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9976  aleph0  9977  alephlim  9978  alephsuc  9979