Definition df-om 3950

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 3951 for an alternate definition. Later, when we assume the Axiom of Infinity, we show om is a set in omex 5733, and om can then be defined per dfom3 5737 (the smallest inductive set) and dfom4 5739.

Note: the natural numbers om are a subset of the ordinal numbers df-on 3661. They are completely different from the natural numbers NN (df-n 7108) 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.

Assertion

Ref	Expression
df-om	|- om = {x | (Ord x /\ 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 3949	. 2 class om
2		vx	. . . . . 6 set x
3	2	cv 1297	. . . . 5 class x
4	3	word 3656	. . . 4 wff Ord x
5		vy	. . . . . . . 8 set y
6	5	cv 1297	. . . . . . 7 class y
7	6	wlim 3658	. . . . . 6 wff Lim y
8	3, 6	wcel 1300	. . . . . 6 wff x e. y
9	7, 8	wi 3	. . . . 5 wff (Lim y -> x e. y)
10	9, 5	wal 1296	. . . 4 wff A.y(Lim y -> x e. y)
11	4, 10	wa 240	. . 3 wff (Ord x /\ A.y(Lim y -> x e. y))
12	11, 2	cab 1871	. 2 class {x | (Ord x /\ A.y(Lim y -> x e. y))}
13	1, 12	wceq 1298	1 wff om = {x | (Ord x /\ A.y(Lim y -> x e. y))}

This definition is referenced by:  dfom2 3951  elom 3952  dfom5 14081

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