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Definition df-aleph 9019
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 9142, alephsuc 9144, and alephlim 9143. 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
StepHypRef Expression
1 cale 9015 . 2 class
2 char 8670 . . 3 class har
3 com 7265 . . 3 class ω
42, 3crdg 7711 . 2 class rec(har, ω)
51, 4wceq 1652 1 wff ℵ = rec(har, ω)
Colors of variables: wff setvar class
This definition is referenced by:  alephfnon  9141  aleph0  9142  alephlim  9143  alephsuc  9144
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