Definition df-aleph 9893

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 10019, alephsuc 10021, and alephlim 10020. 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 9889	. 2 class ℵ
2		char 9509	. . 3 class har
3		com 7842	. . 3 class ω
4	2, 3	crdg 8377	. 2 class rec(har, ω)
5	1, 4	wceq 1540	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10018  aleph0  10019  alephlim  10020  alephsuc  10021