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Definition df-om 7581
 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 7582 for an alternate definition. Later, when we assume the Axiom of Infinity, we show ω is a set in omex 9132, and ω can then be defined per dfom3 9136 (the smallest inductive set) and dfom4 9138. Note: the natural numbers ω are a subset of the ordinal numbers df-on 6174. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-nn 11668) with analogous properties and operations, but they will be different sets. (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 7580 . 2 class ω
2 vy . . . . . . 7 setvar 𝑦
32cv 1538 . . . . . 6 class 𝑦
43wlim 6171 . . . . 5 wff Lim 𝑦
5 vx . . . . . 6 setvar 𝑥
65, 2wel 2113 . . . . 5 wff 𝑥𝑦
74, 6wi 4 . . . 4 wff (Lim 𝑦𝑥𝑦)
87, 2wal 1537 . . 3 wff 𝑦(Lim 𝑦𝑥𝑦)
9 con0 6170 . . 3 class On
108, 5, 9crab 3075 . 2 class {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}
111, 10wceq 1539 1 wff ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦𝑥𝑦)}
 Colors of variables: wff setvar class This definition is referenced by:  dfom2  7582  elom  7583  omsson  7584
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