Definition df-aleph 9976

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 10102, alephsuc 10104, and alephlim 10103. 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 9972	. 2 class ℵ
2		char 9592	. . 3 class har
3		com 7883	. . 3 class ω
4	2, 3	crdg 8445	. 2 class rec(har, ω)
5	1, 4	wceq 1540	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10101  aleph0  10102  alephlim  10103  alephsuc  10104