Definition df-aleph 9698

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 9822, alephsuc 9824, and alephlim 9823. 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 9694	. 2 class ℵ
2		char 9315	. . 3 class har
3		com 7712	. . 3 class ω
4	2, 3	crdg 8240	. 2 class rec(har, ω)
5	1, 4	wceq 1539	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9821  aleph0  9822  alephlim  9823  alephsuc  9824