Definition df-iom 4575

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 4353. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8879) 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 7171. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4576 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 4574	. 2 class ω
2		c0 3414	. . . . . 6 class ∅
3		vx	. . . . . . 7 setvar 𝑥
4	3	cv 1347	. . . . . 6 class 𝑥
5	2, 4	wcel 2141	. . . . 5 wff ∅ ∈ 𝑥
6		vy	. . . . . . . . 9 setvar 𝑦
7	6	cv 1347	. . . . . . . 8 class 𝑦
8	7	csuc 4350	. . . . . . 7 class suc 𝑦
9	8, 4	wcel 2141	. . . . . 6 wff suc 𝑦 ∈ 𝑥
10	9, 6, 4	wral 2448	. . . . 5 wff ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥
11	5, 10	wa 103	. . . 4 wff (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)
12	11, 3	cab 2156	. . 3 class {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
13	12	cint 3831	. 2 class ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
14	1, 13	wceq 1348	1 wff ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}

This definition is referenced by:  dfom3  4576