Definition df-aleph 8976

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 9099, alephsuc 9101, and alephlim 9100. 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 8972	. 2 class ℵ
2		char 8628	. . 3 class har
3		com 7231	. . 3 class ω
4	2, 3	crdg 7675	. 2 class rec(har, ω)
5	1, 4	wceq 1632	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9098  aleph0  9099  alephlim  9100  alephsuc  9101