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Definition df-iom 4511
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 ω are a subset of the ordinal numbers df-on 4296. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8743) 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 7065. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4512 instead for naming consistency with set.mm. (New usage is discouraged.)

Assertion
Ref Expression
df-iom ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-iom
StepHypRef Expression
1 com 4510 . 2 class ω
2 c0 3366 . . . . . 6 class
3 vx . . . . . . 7 setvar 𝑥
43cv 1331 . . . . . 6 class 𝑥
52, 4wcel 1481 . . . . 5 wff ∅ ∈ 𝑥
6 vy . . . . . . . . 9 setvar 𝑦
76cv 1331 . . . . . . . 8 class 𝑦
87csuc 4293 . . . . . . 7 class suc 𝑦
98, 4wcel 1481 . . . . . 6 wff suc 𝑦𝑥
109, 6, 4wral 2417 . . . . 5 wff 𝑦𝑥 suc 𝑦𝑥
115, 10wa 103 . . . 4 wff (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
1211, 3cab 2126 . . 3 class {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
1312cint 3777 . 2 class {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
141, 13wceq 1332 1 wff ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
Colors of variables: wff set class
This definition is referenced by:  dfom3  4512
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