Definition df-aleph 8804

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 8927, alephsuc 8929, and alephlim 8928. 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 8800	. 2 class ℵ
2		char 8502	. . 3 class har
3		com 7107	. . 3 class ω
4	2, 3	crdg 7550	. 2 class rec(har, ω)
5	1, 4	wceq 1523	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  8926  aleph0  8927  alephlim  8928  alephsuc  8929