Theorem nnnn0i 9257

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 9256	. 2 |- ( N e. NN -> N e. NN0 )
3	1, 2	ax-mp 5	1 |- N e. NN0

Syntax hints:   e. wcel 2167   NNcn 8990   NN0cn0 9249

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-n0 9250

This theorem is referenced by:  1nn0  9265  2nn0  9266  3nn0  9267  4nn0  9268  5nn0  9269  6nn0  9270  7nn0  9271  8nn0  9272  9nn0  9273  numlt  9481  declei  9492  numlti  9493  pockthi  12527  dec5dvds2  12582  modxp1i  12587