Definition df-om 2373

Description: Define the class of natural numbers. Our definition is a variant of the Definition of N of [BellMachover] p. 471. See dfom2 2374 for an alternate definition. Later, when we assume the Axiom of Infinity, we show ω is a set in omex 3475, and ω can then be defined per dfom3 3477 (the smallest inductive set) and dfom4 3479. Note: the natural numbers ω are a subset of the ordinal numbers df-on 2203. These are different from the natural number subset of complex numbers defined later (df-n 4423), although the two sets have analogous properties and operations defined on them.

Assertion

Ref	Expression
df-om	⊢ ω = {x∣(Ord x ∧ ∀y(Lim y → x ∈ y))}

Distinct variable group(s):   x,y

Detailed syntax breakdown of Definition df-om

Step	Hyp	Ref	Expression
1		com 2372	. 2 class ω
2		vx	. . . . . 6 set x
3	2	cv 1089	. . . . 5 class x
4	3	word 2198	. . . 4 wff Ord x
5		vy	. . . . . . . 8 set y
6	5	cv 1089	. . . . . . 7 class y
7	6	wlim 2200	. . . . . 6 wff Lim y
8	2, 5	wel 803	. . . . . 6 wff x ∈ y
9	7, 8	wi 2	. . . . 5 wff (Lim y → x ∈ y)
10	9, 5	wal 672	. . . 4 wff ∀y(Lim y → x ∈ y)
11	4, 10	wa 196	. . 3 wff (Ord x ∧ ∀y(Lim y → x ∈ y))
12	11, 2	cab 1090	. 2 class {x∣(Ord x ∧ ∀y(Lim y → x ∈ y))}
13	1, 12	wceq 1091	1 wff ω = {x∣(Ord x ∧ ∀y(Lim y → x ∈ y))}

This definition is referenced by:  dfom2 2374  elom 2375

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