Definition df-aleph 8649

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 8772, alephsuc 8774, and alephlim 8773. 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 8645	. 2 class ℵ
2		char 8344	. . 3 class har
3		com 6957	. . 3 class ω
4	2, 3	crdg 7392	. 2 class rec(har, ω)
5	1, 4	wceq 1475	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  8771  aleph0  8772  alephlim  8773  alephsuc  8774