Definition df-aleph 9369

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 9492, alephsuc 9494, and alephlim 9493. 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 9365	. 2 class ℵ
2		char 9020	. . 3 class har
3		com 7580	. . 3 class ω
4	2, 3	crdg 8045	. 2 class rec(har, ω)
5	1, 4	wceq 1537	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9491  aleph0  9492  alephlim  9493  alephsuc  9494