Definition df-aleph 9839

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 9963, alephsuc 9965, and alephlim 9964. 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 9835	. 2 class ℵ
2		char 9448	. . 3 class har
3		com 7802	. . 3 class ω
4	2, 3	crdg 8334	. 2 class rec(har, ω)
5	1, 4	wceq 1541	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  9962  aleph0  9963  alephlim  9964  alephsuc  9965