Definition df-aleph 9910

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 10034, alephsuc 10036, and alephlim 10035. 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 9906	. 2 class ℵ
2		char 9502	. . 3 class har
3		com 7846	. . 3 class ω
4	2, 3	crdg 8380	. 2 class rec(har, ω)
5	1, 4	wceq 1561	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10033  aleph0  10034  alephlim  10035  alephsuc  10036