Definition df-nn 12147

Description: Define the set of positive integers. Some authors, especially in analysis books, call these the natural numbers, whereas other authors choose to include 0 in their definition of natural numbers. Note that ℕ is a subset of complex numbers (nnsscn 12151), in contrast to the more elementary ordinal natural numbers ω, df-om 7807). See nnind 12164 for the principle of mathematical induction. See df-n0 12403 for the set of nonnegative integers ℕ₀. See dfn2 12415 for ℕ defined in terms of ℕ₀.

This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 9556 in certain proofs. For a more conventional and intuitive definition ("the smallest set of reals containing 1 as well as the successor of every member") see dfnn3 12160 (or its slight variant dfnn2 12159). (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 3-May-2014.)

Assertion

Ref	Expression
df-nn	⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)

Detailed syntax breakdown of Definition df-nn

Step	Hyp	Ref	Expression
1		cn 12146	. 2 class ℕ
2		vx	. . . . 5 setvar 𝑥
3		cvv 3438	. . . . 5 class V
4	2	cv 1539	. . . . . 6 class 𝑥
5		c1 11029	. . . . . 6 class 1
6		caddc 11031	. . . . . 6 class +
7	4, 5, 6	co 7353	. . . . 5 class (𝑥 + 1)
8	2, 3, 7	cmpt 5176	. . . 4 class (𝑥 ∈ V ↦ (𝑥 + 1))
9	8, 5	crdg 8338	. . 3 class rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1)
10		com 7806	. . 3 class ω
11	9, 10	cima 5626	. 2 class (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
12	1, 11	wceq 1540	1 wff ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)

This definition is referenced by:  nnexALT  12148  peano5nni  12149  1nn  12157  peano2nn  12158