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Definition df-nn 11630
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 11634), in contrast to the more elementary ordinal natural numbers ω, df-om 7565). See nnind 11647 for the principle of mathematical induction. See df-n0 11890 for the set of nonnegative integers 0. See dfn2 11902 for defined in terms of 0.

This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 9092 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 11643 (or its slight variant dfnn2 11642). (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 11629 . 2 class
2 vx . . . . 5 setvar 𝑥
3 cvv 3444 . . . . 5 class V
42cv 1537 . . . . . 6 class 𝑥
5 c1 10531 . . . . . 6 class 1
6 caddc 10533 . . . . . 6 class +
74, 5, 6co 7139 . . . . 5 class (𝑥 + 1)
82, 3, 7cmpt 5113 . . . 4 class (𝑥 ∈ V ↦ (𝑥 + 1))
98, 5crdg 8032 . . 3 class rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1)
10 com 7564 . . 3 class ω
119, 10cima 5526 . 2 class (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
121, 11wceq 1538 1 wff ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
Colors of variables: wff setvar class
This definition is referenced by:  nnexALT  11631  peano5nni  11632  1nn  11640  peano2nn  11641
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