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Definition df-iom 4590
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 4368. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8919) 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 7192. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4591 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 4589 . 2  class  om
2 c0 3422 . . . . . 6  class  (/)
3 vx . . . . . . 7  setvar  x
43cv 1352 . . . . . 6  class  x
52, 4wcel 2148 . . . . 5  wff  (/)  e.  x
6 vy . . . . . . . . 9  setvar  y
76cv 1352 . . . . . . . 8  class  y
87csuc 4365 . . . . . . 7  class  suc  y
98, 4wcel 2148 . . . . . 6  wff  suc  y  e.  x
109, 6, 4wral 2455 . . . . 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 2163 . . 3  class  { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
) }
1312cint 3844 . 2  class  |^| { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x
) }
141, 13wceq 1353 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  4591
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