Definition df-aleph 9945

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 10069, alephsuc 10071, and alephlim 10070. 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 9941	. 2 class ℵ
2		char 9528	. . 3 class har
3		com 7862	. . 3 class ω
4	2, 3	crdg 8398	. 2 class rec(har, ω)
5	1, 4	wceq 1570	1 wff ℵ = rec(har, ω)

This definition is used by:  alephfnon  10068  aleph0  10069  alephlim  10070  alephsuc  10071