MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-nn Unicode version

Definition df-nn 9763
Description: The natural numbers of analysis start at one (unlike the ordinal natural numbers, i.e. the members of the set  om, df-om 4673, 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 9780 for the principle of mathematical induction. See dfnn2 9775 for a slight variant. See df-n0 9982 for the set of nonnegative integers  NN0 starting at zero. See dfn2 9994 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 9776. (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 9762 . 2  class  NN
2 vx . . . . 5  set  x
3 cvv 2801 . . . . 5  class  _V
42cv 1631 . . . . . 6  class  x
5 c1 8754 . . . . . 6  class  1
6 caddc 8756 . . . . . 6  class  +
74, 5, 6co 5874 . . . . 5  class  ( x  +  1 )
82, 3, 7cmpt 4093 . . . 4  class  ( x  e.  _V  |->  ( x  +  1 ) )
98, 5crdg 6438 . . 3  class  rec (
( x  e.  _V  |->  ( x  +  1
) ) ,  1 )
10 com 4672 . . 3  class  om
119, 10cima 4708 . 2  class  ( rec ( ( x  e. 
_V  |->  ( x  + 
1 ) ) ,  1 ) " om )
121, 11wceq 1632 1  wff  NN  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  1 ) " om )
Colors of variables: wff set class
This definition is referenced by:  nnexALT  9764  peano5nni  9765  1nn  9773  peano2nn  9774
  Copyright terms: Public domain W3C validator