Theorem nnnn0i 9305

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 9304	. 2 |- ( N e. NN -> N e. NN0 )
3	1, 2	ax-mp 5	1 |- N e. NN0

Syntax hints:   e. wcel 2176   NNcn 9038   NN0cn0 9297

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-n0 9298

This theorem is referenced by:  1nn0  9313  2nn0  9314  3nn0  9315  4nn0  9316  5nn0  9317  6nn0  9318  7nn0  9319  8nn0  9320  9nn0  9321  numlt  9530  declei  9541  numlti  9542  pockthi  12714  dec5dvds2  12769  modxp1i  12774