Definition df-iom 4695

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 4471. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 9203) 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 7465. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4696 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 4694	. 2 class om
2		c0 3496	. . . . . 6 class (/)
3		vx	. . . . . . 7 setvar x
4	3	cv 1397	. . . . . 6 class x
5	2, 4	wcel 2202	. . . . 5 wff (/) e. x
6		vy	. . . . . . . . 9 setvar y
7	6	cv 1397	. . . . . . . 8 class y
8	7	csuc 4468	. . . . . . 7 class suc y
9	8, 4	wcel 2202	. . . . . 6 wff suc y e. x
10	9, 6, 4	wral 2511	. . . . 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 2217	. . 3 class { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
13	12	cint 3933	. 2 class |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
14	1, 13	wceq 1398	1 wff om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }

This definition is referenced by:  dfom3  4696