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Definition df-iom 4608
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 4386. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8951) with analogous properties and operations, but they will be different sets.

We are unable to use the terms finite ordinal and natural number interchangeably, as shown at exmidonfin 7224. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4609 instead for naming consistency with set.mm. (New usage is discouraged.)

Assertion
Ref Expression
df-iom  |-  om  =  |^| { x  |  (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x ) }
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-iom
StepHypRef Expression
1 com 4607 . 2  class  om
2 c0 3437 . . . . . 6  class  (/)
3 vx . . . . . . 7  setvar  x
43cv 1363 . . . . . 6  class  x
52, 4wcel 2160 . . . . 5  wff  (/)  e.  x
6 vy . . . . . . . . 9  setvar  y
76cv 1363 . . . . . . . 8  class  y
87csuc 4383 . . . . . . 7  class  suc  y
98, 4wcel 2160 . . . . . 6  wff  suc  y  e.  x
109, 6, 4wral 2468 . . . . 5  wff  A. y  e.  x  suc  y  e.  x
115, 10wa 104 . . . 4  wff  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
1211, 3cab 2175 . . 3  class  { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
) }
1312cint 3859 . 2  class  |^| { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
) }
141, 13wceq 1364 1  wff  om  =  |^| { x  |  (
(/)  e.  x  /\  A. y  e.  x  suc  y  e.  x ) }
Colors of variables: wff set class
This definition is referenced by:  dfom3  4609
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