Definition df-aleph 10021

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 10145, alephsuc 10147, and alephlim 10146. 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 10017	. 2 class ℵ
2		char 9550	. . 3 class har
3		com 7877	. . 3 class ω
4	2, 3	crdg 8417	. 2 class rec(har, ω)
5	1, 4	wceq 1569	1 wff ℵ = rec(har, ω)

This definition is used by:  alephfnon  10144  aleph0  10145  alephlim  10146  alephsuc  10147