Definition df-aleph 9864

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 9988, alephsuc 9990, and alephlim 9989. 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 9860	. 2 class ℵ
2		char 9471	. . 3 class har
3		com 7817	. . 3 class ω
4	2, 3	crdg 8348	. 2 class rec(har, ω)
5	1, 4	wceq 1542	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9987  aleph0  9988  alephlim  9989  alephsuc  9990