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Definition df-nn 9747
Description: The natural numbers of analysis start at one (unlike the ordinal natural numbers, i.e. the members of the set  om, df-om 4657, which start at zero). This is the convention used by most analysis books, and it is often convenient in proofs because we don't have to worry about division by zero. See nnind 9764 for the principle of mathematical induction. See dfnn2 9759 for a slight variant. See df-n0 9966 for the set of nonnegative integers  NN0 starting at zero. See dfn2 9978 for  NN defined in terms of  NN0.

This is a technical definition that helps us avoid the Axiom of Infinity 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 9760. (Contributed by NM, 10-Jan-1997.)

Assertion
Ref Expression
df-nn  |-  NN  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  1 ) " om )

Detailed syntax breakdown of Definition df-nn
StepHypRef Expression
1 cn 9746 . 2  class  NN
2 vx . . . . 5  set  x
3 cvv 2788 . . . . 5  class  _V
42cv 1622 . . . . . 6  class  x
5 c1 8738 . . . . . 6  class  1
6 caddc 8740 . . . . . 6  class  +
74, 5, 6co 5858 . . . . 5  class  ( x  +  1 )
82, 3, 7cmpt 4077 . . . 4  class  ( x  e.  _V  |->  ( x  +  1 ) )
98, 5crdg 6422 . . 3  class  rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  1 )
10 com 4656 . . 3  class  om
119, 10cima 4692 . 2  class  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  1 ) " om )
121, 11wceq 1623 1  wff  NN  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  1 ) " om )
Colors of variables: wff set class
This definition is referenced by:  nnexALT  9748  peano5nni  9749  1nn  9757  peano2nn  9758
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