Definition df-aleph 9934

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 10060, alephsuc 10062, and alephlim 10061. 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 9930	. 2 class ℵ
2		char 9550	. . 3 class har
3		com 7854	. . 3 class ω
4	2, 3	crdg 8408	. 2 class rec(har, ω)
5	1, 4	wceq 1541	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10059  aleph0  10060  alephlim  10061  alephsuc  10062