Definition df-aleph 9353

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 9477, alephsuc 9479, and alephlim 9478. 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 9349	. 2 class ℵ
2		char 9004	. . 3 class har
3		com 7560	. . 3 class ω
4	2, 3	crdg 8028	. 2 class rec(har, ω)
5	1, 4	wceq 1538	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9476  aleph0  9477  alephlim  9478  alephsuc  9479