Definition df-aleph 9825

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 9949, alephsuc 9951, and alephlim 9950. 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 9821	. 2 class ℵ
2		char 9437	. . 3 class har
3		com 7791	. . 3 class ω
4	2, 3	crdg 8323	. 2 class rec(har, ω)
5	1, 4	wceq 1541	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9948  aleph0  9949  alephlim  9950  alephsuc  9951