Theorem nnnn0i 9113

Description: A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)

Hypothesis

Ref	Expression
nnnn0.1	|- N e. NN

Assertion

Ref	Expression
nnnn0i	|- N e. NN0

Proof of Theorem nnnn0i

Step	Hyp	Ref	Expression
1		nnnn0.1	. 2 |- N e. NN
2		nnnn0 9112	. 2 |- ( N e. NN -> N e. NN0 )
3	1, 2	ax-mp 5	1 |- N e. NN0

Syntax hints:   e. wcel 2135   NNcn 8848   NN0cn0 9105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-n0 9106

This theorem is referenced by:  1nn0  9121  2nn0  9122  3nn0  9123  4nn0  9124  5nn0  9125  6nn0  9126  7nn0  9127  8nn0  9128  9nn0  9129  numlt  9337  declei  9348  numlti  9349