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Definition df-nn 10408
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 10412), in contrast to the more elementary ordinal natural numbers , df-om 6561). See nnind 10425 for the principle of mathematical induction. See dfnn2 10420 for a slight variant. See df-n0 10665 for the set of nonnegative integers . See dfn2 10677 for defined in terms of .

This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 7932 in certain proofs. For a more conventional and intuitive definition ("the smallest set of reals containing as well as the successor of every member") see dfnn3 10421. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 3-May-2014.)

Assertion
Ref Expression
df-nn

Detailed syntax breakdown of Definition df-nn
StepHypRef Expression
1 cn 10407 . 2
2 vx . . . . 5
3 cvv 3052 . . . . 5
42cv 1369 . . . . . 6
5 c1 9368 . . . . . 6
6 caddc 9370 . . . . . 6
74, 5, 6co 6174 . . . . 5
82, 3, 7cmpt 4432 . . . 4
98, 5crdg 6949 . . 3
10 com 6560 . . 3
119, 10cima 4925 . 2
121, 11wceq 1370 1
Colors of variables: wff setvar class
This definition is referenced by:  nnexALT  10409  peano5nni  10410  1nn  10418  peano2nn  10419
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