Definition df-iom 4608

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 om are a subset of the ordinal numbers df-on 4386. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8951) 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 7224. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4609 instead for naming consistency with set.mm. (New usage is discouraged.)

Assertion

Ref	Expression
df-iom	|- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-iom

Step	Hyp	Ref	Expression
1		com 4607	. 2 class om
2		c0 3437	. . . . . 6 class (/)
3		vx	. . . . . . 7 setvar x
4	3	cv 1363	. . . . . 6 class x
5	2, 4	wcel 2160	. . . . 5 wff (/) e. x
6		vy	. . . . . . . . 9 setvar y
7	6	cv 1363	. . . . . . . 8 class y
8	7	csuc 4383	. . . . . . 7 class suc y
9	8, 4	wcel 2160	. . . . . 6 wff suc y e. x
10	9, 6, 4	wral 2468	. . . . 5 wff A. y e. x suc y e. x
11	5, 10	wa 104	. . . 4 wff ( (/) e. x /\ A. y e. x suc y e. x )
12	11, 3	cab 2175	. . 3 class { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
13	12	cint 3859	. 2 class |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
14	1, 13	wceq 1364	1 wff om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }

This definition is referenced by:  dfom3  4609