ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elznn0nn Unicode version

Theorem elznn0nn 8762
Description: Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
Assertion
Ref Expression
elznn0nn  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )

Proof of Theorem elznn0nn
StepHypRef Expression
1 elz 8750 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
2 andi 767 . . 3  |-  ( ( N  e.  RR  /\  ( ( N  =  0  \/  N  e.  NN )  \/  -u N  e.  NN ) )  <->  ( ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN )
)  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
3 df-3or 925 . . . 4  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( ( N  =  0  \/  N  e.  NN )  \/  -u N  e.  NN ) )
43anbi2i 445 . . 3  |-  ( ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) )  <->  ( N  e.  RR  /\  ( ( N  =  0  \/  N  e.  NN )  \/  -u N  e.  NN ) ) )
5 nn0re 8680 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  RR )
65pm4.71ri 384 . . . . 5  |-  ( N  e.  NN0  <->  ( N  e.  RR  /\  N  e. 
NN0 ) )
7 elnn0 8673 . . . . . . 7  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
8 orcom 682 . . . . . . 7  |-  ( ( N  e.  NN  \/  N  =  0 )  <-> 
( N  =  0  \/  N  e.  NN ) )
97, 8bitri 182 . . . . . 6  |-  ( N  e.  NN0  <->  ( N  =  0  \/  N  e.  NN ) )
109anbi2i 445 . . . . 5  |-  ( ( N  e.  RR  /\  N  e.  NN0 )  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN ) ) )
116, 10bitri 182 . . . 4  |-  ( N  e.  NN0  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN ) ) )
1211orbi1i 715 . . 3  |-  ( ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) )  <->  ( ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN )
)  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
132, 4, 123bitr4i 210 . 2  |-  ( ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) )  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
141, 13bitri 182 1  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    \/ wo 664    \/ w3o 923    = wceq 1289    e. wcel 1438   RRcr 7347   0cc0 7348   -ucneg 7652   NNcn 8420   NN0cn0 8671   ZZcz 8748
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-i2m1 7448  ax-rnegex 7452
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655  df-neg 7654  df-inn 8421  df-n0 8672  df-z 8749
This theorem is referenced by:  peano2z  8784  zindd  8862  expcl2lemap  9963  mulexpzap  9991  expaddzap  9995  expmulzap  9997  absexpzap  10509
  Copyright terms: Public domain W3C validator