Definition df-aleph 10003

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 10129, alephsuc 10131, and alephlim 10130. 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 9999	. 2 class ℵ
2		char 9619	. . 3 class har
3		com 7897	. . 3 class ω
4	2, 3	crdg 8459	. 2 class rec(har, ω)
5	1, 4	wceq 1537	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10128  aleph0  10129  alephlim  10130  alephsuc  10131