Definition df-aleph 9899

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 10025, alephsuc 10027, and alephlim 10026. 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 9895	. 2 class ℵ
2		char 9515	. . 3 class har
3		com 7844	. . 3 class ω
4	2, 3	crdg 8379	. 2 class rec(har, ω)
5	1, 4	wceq 1540	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10024  aleph0  10025  alephlim  10026  alephsuc  10027