Definition df-om 7066

Description: Define the class of natural numbers, which are all ordinal numbers that are less than every limit ordinal, i.e. all finite ordinals. Our definition is a variant of the Definition of N of [BellMachover] p. 471. See dfom2 7067 for an alternate definition. Later, when we assume the Axiom of Infinity, we show om is a set in omex 8540, and om can then be defined per dfom3 8544 (the smallest inductive set) and dfom4 8546.

Note: the natural numbers om are a subset of the ordinal numbers df-on 5727. They are completely different from the natural numbers NN (df-nn 11021) that are a subset of the complex numbers defined much later in our development, although the two sets have analogous properties and operations defined on them. (Contributed by NM, 15-May-1994.)

Assertion

Ref	Expression
df-om	|- om = { x e. On | A. y ( Lim y -> x e. y ) }

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-om

Step	Hyp	Ref	Expression
1		com 7065	. 2 class om
2		vy	. . . . . . 7 setvar y
3	2	cv 1482	. . . . . 6 class y
4	3	wlim 5724	. . . . 5 wff Lim y
5		vx	. . . . . 6 setvar x
6	5, 2	wel 1991	. . . . 5 wff x e. y
7	4, 6	wi 4	. . . 4 wff ( Lim y -> x e. y )
8	7, 2	wal 1481	. . 3 wff A. y ( Lim y -> x e. y )
9		con0 5723	. . 3 class On
10	8, 5, 9	crab 2916	. 2 class { x e. On | A. y ( Lim y -> x e. y ) }
11	1, 10	wceq 1483	1 wff om = { x e. On | A. y ( Lim y -> x e. y ) }

This definition is referenced by:  dfom2  7067  elom  7068