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Definition df-om 7028
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 7029 for an alternate definition. Later, when we assume the Axiom of Infinity, we show ω is a set in omex 8500, and ω can then be defined per dfom3 8504 (the smallest inductive set) and dfom4 8506.

Note: the natural numbers ω are a subset of the ordinal numbers df-on 5696. They are completely different from the natural numbers (df-nn 10981) 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 ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-om
StepHypRef Expression
1 com 7027 . 2 class ω
2 vy . . . . . . 7 setvar 𝑦
32cv 1479 . . . . . 6 class 𝑦
43wlim 5693 . . . . 5 wff Lim 𝑦
5 vx . . . . . 6 setvar 𝑥
65, 2wel 1988 . . . . 5 wff 𝑥𝑦
74, 6wi 4 . . . 4 wff (Lim 𝑦𝑥𝑦)
87, 2wal 1478 . . 3 wff 𝑦(Lim 𝑦𝑥𝑦)
9 con0 5692 . . 3 class On
108, 5, 9crab 2912 . 2 class {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}
111, 10wceq 1480 1 wff ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}
Colors of variables: wff setvar class
This definition is referenced by:  dfom2  7029  elom  7030
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