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Definition df-om 6701
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 6702 for an alternate definition. Later, when we assume the Axiom of Infinity, we show is a set in omex 8081, and can then be defined per dfom3 8085 (the smallest inductive set) and dfom4 8087.

Note: the natural numbers are a subset of the ordinal numbers df-on 4887. They are completely different from the natural numbers (df-nn 10562) 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
Distinct variable group:   ,

Detailed syntax breakdown of Definition df-om
StepHypRef Expression
1 com 6700 . 2
2 vy . . . . . . 7
32cv 1394 . . . . . 6
43wlim 4884 . . . . 5
5 vx . . . . . 6
65, 2wel 1819 . . . . 5
74, 6wi 4 . . . 4
87, 2wal 1393 . . 3
9 con0 4883 . . 3
108, 5, 9crab 2811 . 2
111, 10wceq 1395 1
Colors of variables: wff setvar class
This definition is referenced by:  dfom2  6702  elom  6703
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