Definition df-aleph 9971

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 10097, alephsuc 10099, and alephlim 10098. 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 9967	. 2 class ℵ
2		char 9587	. . 3 class har
3		com 7880	. . 3 class ω
4	2, 3	crdg 8442	. 2 class rec(har, ω)
5	1, 4	wceq 1535	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10096  aleph0  10097  alephlim  10098  alephsuc  10099