Definition df-iom 4657

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 4433. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 9072) 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 7333. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4658 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 4656	. 2 class ω
2		c0 3468	. . . . . 6 class ∅
3		vx	. . . . . . 7 setvar 𝑥
4	3	cv 1372	. . . . . 6 class 𝑥
5	2, 4	wcel 2178	. . . . 5 wff ∅ ∈ 𝑥
6		vy	. . . . . . . . 9 setvar 𝑦
7	6	cv 1372	. . . . . . . 8 class 𝑦
8	7	csuc 4430	. . . . . . 7 class suc 𝑦
9	8, 4	wcel 2178	. . . . . 6 wff suc 𝑦 ∈ 𝑥
10	9, 6, 4	wral 2486	. . . . 5 wff ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥
11	5, 10	wa 104	. . . 4 wff (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)
12	11, 3	cab 2193	. . 3 class {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
13	12	cint 3899	. 2 class ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
14	1, 13	wceq 1373	1 wff ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}

This definition is referenced by:  dfom3  4658