Definition df-iom 4738

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 4513. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 9305) 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 7546. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4739 instead for naming consistency with set.mm. (New usage is discouraged.)

Assertion

Ref	Expression
df-iom	⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-iom

Step	Hyp	Ref	Expression
1		com 4737	. 2 class ω
2		c0 3520	. . . . . 6 class ∅
3		vx	. . . . . . 7 setvar 𝑥
4	3	cv 1401	. . . . . 6 class 𝑥
5	2, 4	wcel 2209	. . . . 5 wff ∅ ∈ 𝑥
6		vy	. . . . . . . . 9 setvar 𝑦
7	6	cv 1401	. . . . . . . 8 class 𝑦
8	7	csuc 4510	. . . . . . 7 class suc 𝑦
9	8, 4	wcel 2209	. . . . . 6 wff suc 𝑦 ∈ 𝑥
10	9, 6, 4	wral 2528	. . . . 5 wff ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥
11	5, 10	wa 104	. . . 4 wff (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)
12	11, 3	cab 2224	. . 3 class {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
13	12	cint 3970	. 2 class ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
14	1, 13	wceq 1402	1 wff ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}

This definition is used by:  dfom3  4739