Definition df-aleph 9925

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 10049, alephsuc 10051, and alephlim 10050. 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 9921	. 2 class ℵ
2		char 9517	. . 3 class har
3		com 7861	. . 3 class ω
4	2, 3	crdg 8395	. 2 class rec(har, ω)
5	1, 4	wceq 1568	1 wff ℵ = rec(har, ω)

This definition is referenced by:  alephfnon  10048  aleph0  10049  alephlim  10050  alephsuc  10051