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Definition df-nn 9834
Description: The natural numbers of analysis start at one (unlike the ordinal natural numbers, i.e. the members of the set  om, df-om 4736, 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 9851 for the principle of mathematical induction. See dfnn2 9846 for a slight variant. See df-n0 10055 for the set of nonnegative integers  NN0 starting at zero. See dfn2 10067 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 9847. (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 9833 . 2  class  NN
2 vx . . . . 5  set  x
3 cvv 2864 . . . . 5  class  _V
42cv 1641 . . . . . 6  class  x
5 c1 8825 . . . . . 6  class  1
6 caddc 8827 . . . . . 6  class  +
74, 5, 6co 5942 . . . . 5  class  ( x  +  1 )
82, 3, 7cmpt 4156 . . . 4  class  ( x  e.  _V  |->  ( x  +  1 ) )
98, 5crdg 6506 . . 3  class  rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  1 )
10 com 4735 . . 3  class  om
119, 10cima 4771 . 2  class  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  1 ) " om )
121, 11wceq 1642 1  wff  NN  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  1 ) " om )
Colors of variables: wff set class
This definition is referenced by:  nnexALT  9835  peano5nni  9836  1nn  9844  peano2nn  9845
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