Definition df-aleph 9372

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 9495, alephsuc 9497, and alephlim 9496. 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 9368	. 2 class ℵ
2		char 9023	. . 3 class har
3		com 7583	. . 3 class ω
4	2, 3	crdg 8048	. 2 class rec(har, ω)
5	1, 4	wceq 1536	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9494  aleph0  9495  alephlim  9496  alephsuc  9497