Definition df-om 7581

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 7582 for an alternate definition. Later, when we assume the Axiom of Infinity, we show ω is a set in omex 9106, and ω can then be defined per dfom3 9110 (the smallest inductive set) and dfom4 9112.

Note: the natural numbers ω are a subset of the ordinal numbers df-on 6195. They are completely different from the natural numbers ℕ (df-nn 11639) 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	⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-om

Step	Hyp	Ref	Expression
1		com 7580	. 2 class ω
2		vy	. . . . . . 7 setvar 𝑦
3	2	cv 1536	. . . . . 6 class 𝑦
4	3	wlim 6192	. . . . 5 wff Lim 𝑦
5		vx	. . . . . 6 setvar 𝑥
6	5, 2	wel 2115	. . . . 5 wff 𝑥 ∈ 𝑦
7	4, 6	wi 4	. . . 4 wff (Lim 𝑦 → 𝑥 ∈ 𝑦)
8	7, 2	wal 1535	. . 3 wff ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)
9		con0 6191	. . 3 class On
10	8, 5, 9	crab 3142	. 2 class {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
11	1, 10	wceq 1537	1 wff ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}

This definition is referenced by:  dfom2  7582  elom  7583