Definition df-aleph 9884

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 10008, alephsuc 10010, and alephlim 10009. 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 9880	. 2 class ℵ
2		char 9490	. . 3 class har
3		com 7831	. . 3 class ω
4	2, 3	crdg 8364	. 2 class rec(har, ω)
5	1, 4	wceq 1550	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10007  aleph0  10008  alephlim  10009  alephsuc  10010