Definition df-aleph 8751

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 8874, alephsuc 8876, and alephlim 8875. 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 8747	. 2 class ℵ
2		char 8446	. . 3 class har
3		com 7050	. . 3 class ω
4	2, 3	crdg 7490	. 2 class rec(har, ω)
5	1, 4	wceq 1481	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  8873  aleph0  8874  alephlim  8875  alephsuc  8876