Definition df-aleph 8766

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 8889, alephsuc 8891, and alephlim 8890. 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 8762	. 2 class ℵ
2		char 8461	. . 3 class har
3		com 7065	. . 3 class ω
4	2, 3	crdg 7505	. 2 class rec(har, ω)
5	1, 4	wceq 1483	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  8888  aleph0  8889  alephlim  8890  alephsuc  8891