Definition df-aleph 9937

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 10063, alephsuc 10065, and alephlim 10064. 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 9933	. 2 class ℵ
2		char 9553	. . 3 class har
3		com 7852	. . 3 class ω
4	2, 3	crdg 8410	. 2 class rec(har, ω)
5	1, 4	wceq 1533	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10062  aleph0  10063  alephlim  10064  alephsuc  10065