Metamath Proof Explorer


Definition df-nn

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 NN is a subset of complex numbers ( nnsscn ), in contrast to the more elementary ordinal natural numbers _om , df-om ). See nnind for the principle of mathematical induction. See df-n0 for the set of nonnegative integers NN0 . See dfn2 for NN defined in terms of NN0 .

This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 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 (or its slight variant dfnn2 ). (Contributed by NM, 10-Jan-1997) (Revised by Mario Carneiro, 3-May-2014)

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

Detailed syntax breakdown

Step Hyp Ref Expression
0 cn
 |-  NN
1 vx
 |-  x
2 cvv
 |-  _V
3 1 cv
 |-  x
4 caddc
 |-  +
5 c1
 |-  1
6 3 5 4 co
 |-  ( x + 1 )
7 1 2 6 cmpt
 |-  ( x e. _V |-> ( x + 1 ) )
8 7 5 crdg
 |-  rec ( ( x e. _V |-> ( x + 1 ) ) , 1 )
9 com
 |-  _om
10 8 9 cima
 |-  ( rec ( ( x e. _V |-> ( x + 1 ) ) , 1 ) " _om )
11 0 10 wceq
 |-  NN = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 1 ) " _om )