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Theorem elnn0 8241
Description: Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
Assertion
Ref Expression
elnn0  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )

Proof of Theorem elnn0
StepHypRef Expression
1 df-n0 8240 . . 3  |-  NN0  =  ( NN  u.  { 0 } )
21eleq2i 2120 . 2  |-  ( A  e.  NN0  <->  A  e.  ( NN  u.  { 0 } ) )
3 elun 3112 . 2  |-  ( A  e.  ( NN  u.  { 0 } )  <->  ( A  e.  NN  \/  A  e. 
{ 0 } ) )
4 c0ex 7079 . . . 4  |-  0  e.  _V
54elsn2 3433 . . 3  |-  ( A  e.  { 0 }  <-> 
A  =  0 )
65orbi2i 689 . 2  |-  ( ( A  e.  NN  \/  A  e.  { 0 } )  <->  ( A  e.  NN  \/  A  =  0 ) )
72, 3, 63bitri 199 1  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 102    \/ wo 639    = wceq 1259    e. wcel 1409    u. cun 2943   {csn 3403   0cc0 6947   NNcn 7990   NN0cn0 8239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-mulcl 7040  ax-i2m1 7047
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-n0 8240
This theorem is referenced by:  0nn0  8254  nn0ge0  8264  nnnn0addcl  8269  nnm1nn0  8280  elnnnn0b  8283  elnn0z  8315  elznn0nn  8316  elznn0  8317  elznn  8318  nn0ind-raph  8414  nn0ledivnn  8785  expp1  9427  expnegap0  9428  expcllem  9431  facp1  9598  faclbnd  9609  faclbnd3  9611  bcn1  9626  ibcval5  9631  nn0enne  10214  nn0o1gt2  10217
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