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Definition df-om 6600
Description: Define the class of natural numbers, which are all ordinal numbers that are less than every limit ordinal, i.e. all finite ordinals. Our definition is a variant of the Definition of N of [BellMachover] p. 471. See dfom2 6601 for an alternate definition. Later, when we assume the Axiom of Infinity, we show  om is a set in omex 7974, and  om can then be defined per dfom3 7978 (the smallest inductive set) and dfom4 7980.

Note: the natural numbers  om are a subset of the ordinal numbers df-on 4796. They are completely different from the natural numbers  NN (df-nn 10453) that are a subset of the complex numbers defined much later in our development, although the two sets have analogous properties and operations defined on them. (Contributed by NM, 15-May-1994.)

Assertion
Ref Expression
df-om  |-  om  =  { x  e.  On  |  A. y ( Lim  y  ->  x  e.  y ) }
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-om
StepHypRef Expression
1 com 6599 . 2  class  om
2 vy . . . . . . 7  setvar  y
32cv 1398 . . . . . 6  class  y
43wlim 4793 . . . . 5  wff  Lim  y
5 vx . . . . . 6  setvar  x
65, 2wel 1827 . . . . 5  wff  x  e.  y
74, 6wi 4 . . . 4  wff  ( Lim  y  ->  x  e.  y )
87, 2wal 1397 . . 3  wff  A. y
( Lim  y  ->  x  e.  y )
9 con0 4792 . . 3  class  On
108, 5, 9crab 2736 . 2  class  { x  e.  On  |  A. y
( Lim  y  ->  x  e.  y ) }
111, 10wceq 1399 1  wff  om  =  { x  e.  On  |  A. y ( Lim  y  ->  x  e.  y ) }
Colors of variables: wff setvar class
This definition is referenced by:  dfom2  6601  elom  6602
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