Definition df-aleph 9933

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 10057, alephsuc 10059, and alephlim 10058. 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 9929	. 2 class ℵ
2		char 9516	. . 3 class har
3		com 7860	. . 3 class ω
4	2, 3	crdg 8394	. 2 class rec(har, ω)
5	1, 4	wceq 1569	1 wff ℵ = rec(har, ω)

This definition is used by:  alephfnon  10056  aleph0  10057  alephlim  10058  alephsuc  10059