Definition df-aleph 10048

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 10172, alephsuc 10174, and alephlim 10173. 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 10044	. 2 class ℵ
2		char 9577	. . 3 class har
3		com 7901	. . 3 class ω
4	2, 3	crdg 8445	. 2 class rec(har, ω)
5	1, 4	wceq 1570	1 wff ℵ = rec(har, ω)

This definition is used by:  alephfnon  10171  aleph0  10172  alephlim  10173  alephsuc  10174