Theorem nnnn0i 9198

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 9197	. 2 |- ( N e. NN -> N e. NN0 )
3	1, 2	ax-mp 5	1 |- N e. NN0

Syntax hints:   e. wcel 2158   NNcn 8933   NN0cn0 9190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-n0 9191

This theorem is referenced by:  1nn0  9206  2nn0  9207  3nn0  9208  4nn0  9209  5nn0  9210  6nn0  9211  7nn0  9212  8nn0  9213  9nn0  9214  numlt  9422  declei  9433  numlti  9434  pockthi  12370