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Definition df-iom 4604
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 4382. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8937) 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 7210. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4605 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 4603 . 2 class ω
2 c0 3436 . . . . . 6 class
3 vx . . . . . . 7 setvar 𝑥
43cv 1362 . . . . . 6 class 𝑥
52, 4wcel 2159 . . . . 5 wff ∅ ∈ 𝑥
6 vy . . . . . . . . 9 setvar 𝑦
76cv 1362 . . . . . . . 8 class 𝑦
87csuc 4379 . . . . . . 7 class suc 𝑦
98, 4wcel 2159 . . . . . 6 wff suc 𝑦𝑥
109, 6, 4wral 2467 . . . . 5 wff 𝑦𝑥 suc 𝑦𝑥
115, 10wa 104 . . . 4 wff (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
1211, 3cab 2174 . . 3 class {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
1312cint 3858 . 2 class {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
141, 13wceq 1363 1 wff ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
Colors of variables: wff set class
This definition is referenced by:  dfom3  4605
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