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Definition df-iom 4257
Description: Define the class of natural numbers as the smallest inductive set, which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82.

Note: the natural numbers  om are a subset of the ordinal numbers df-on 4071. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers with analogous properties and operations, but they will be different sets. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4258 instead for naming consistency with set.mm. (New usage is discouraged.)

Assertion
Ref Expression
df-iom  om  |^|
{  |  (/)  suc  }
Distinct variable group:   ,

Detailed syntax breakdown of Definition df-iom
StepHypRef Expression
1 com 4256 . 2  om
2 c0 3218 . . . . . 6  (/)
3 vx . . . . . . 7  setvar
43cv 1241 . . . . . 6
52, 4wcel 1390 . . . . 5  (/)
6 vy . . . . . . . . 9  setvar
76cv 1241 . . . . . . . 8
87csuc 4068 . . . . . . 7  suc
98, 4wcel 1390 . . . . . 6  suc
109, 6, 4wral 2300 . . . . 5  suc
115, 10wa 97 . . . 4  (/)  suc
1211, 3cab 2023 . . 3  {  |  (/)  suc  }
1312cint 3606 . 2  |^| {  |  (/)  suc  }
141, 13wceq 1242 1  om  |^|
{  |  (/)  suc  }
Colors of variables: wff set class
This definition is referenced by:  dfom3  4258
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