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Definition df-nn 11059
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 is a subset of complex numbers (nnsscn 11063), in contrast to the more elementary ordinal natural numbers ω, df-om 7108). See nnind 11076 for the principle of mathematical induction. See df-n0 11331 for the set of nonnegative integers 0. See dfn2 11343 for defined in terms of 0.

This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 8576 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 11072 (or its slight variant dfnn2 11071). (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
StepHypRef Expression
1 cn 11058 . 2 class
2 vx . . . . 5 setvar 𝑥
3 cvv 3231 . . . . 5 class V
42cv 1522 . . . . . 6 class 𝑥
5 c1 9975 . . . . . 6 class 1
6 caddc 9977 . . . . . 6 class +
74, 5, 6co 6690 . . . . 5 class (𝑥 + 1)
82, 3, 7cmpt 4762 . . . 4 class (𝑥 ∈ V ↦ (𝑥 + 1))
98, 5crdg 7550 . . 3 class rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1)
10 com 7107 . . 3 class ω
119, 10cima 5146 . 2 class (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
121, 11wceq 1523 1 wff ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
Colors of variables: wff setvar class
This definition is referenced by:  nnexALT  11060  peano5nni  11061  1nn  11069  peano2nn  11070
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