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Theorem nnnn0d 8408
Description: A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
Hypothesis
Ref Expression
nnnn0d.1  |-  ( ph  ->  A  e.  NN )
Assertion
Ref Expression
nnnn0d  |-  ( ph  ->  A  e.  NN0 )

Proof of Theorem nnnn0d
StepHypRef Expression
1 nnssnn0 8358 . 2  |-  NN  C_  NN0
2 nnnn0d.1 . 2  |-  ( ph  ->  A  e.  NN )
31, 2sseldi 2998 1  |-  ( ph  ->  A  e.  NN0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   NNcn 8106   NN0cn0 8355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-n0 8356
This theorem is referenced by:  nn0ge2m1nn0  8416  nnzd  8549  eluzge2nn0  8739  modsumfzodifsn  9478  addmodlteq  9480  expinnval  9576  expgt1  9611  expaddzaplem  9616  expaddzap  9617  expmulzap  9619  expnbnd  9693  facwordi  9764  faclbnd  9765  facavg  9770  bcm1k  9784  ibcval5  9787  1elfz0size  9830  resqrexlemnm  10042  resqrexlemcvg  10043  dvdsfac  10405  divalglemnqt  10464  divalglemeunn  10465  gcdval  10495  gcdcl  10502  mulgcd  10549  rplpwr  10560  rppwr  10561  lcmcl  10598  lcmgcdnn  10608  nprmdvds1  10665  rpexp  10676  pw2dvdslemn  10687  sqpweven  10697  2sqpwodd  10698  nn0sqrtelqelz  10728
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