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

Theorem elznn 9028
Description: Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)
Assertion
Ref Expression
elznn  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  -u N  e.  NN0 ) ) )

Proof of Theorem elznn
StepHypRef Expression
1 elz 9014 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
2 recn 7721 . . . . . . . 8  |-  ( N  e.  RR  ->  N  e.  CC )
32negeq0d 8033 . . . . . . 7  |-  ( N  e.  RR  ->  ( N  =  0  <->  -u N  =  0 ) )
43orbi2d 764 . . . . . 6  |-  ( N  e.  RR  ->  (
( -u N  e.  NN  \/  N  =  0
)  <->  ( -u N  e.  NN  \/  -u N  =  0 ) ) )
5 elnn0 8937 . . . . . 6  |-  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  -u N  =  0 ) )
64, 5syl6rbbr 198 . . . . 5  |-  ( N  e.  RR  ->  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  N  =  0 ) ) )
76orbi2d 764 . . . 4  |-  ( N  e.  RR  ->  (
( N  e.  NN  \/  -u N  e.  NN0 ) 
<->  ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) ) )
8 3orrot 953 . . . . 5  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  e.  NN  \/  -u N  e.  NN  \/  N  =  0
) )
9 3orass 950 . . . . 5  |-  ( ( N  e.  NN  \/  -u N  e.  NN  \/  N  =  0 )  <-> 
( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )
108, 9bitri 183 . . . 4  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )
117, 10syl6rbbr 198 . . 3  |-  ( N  e.  RR  ->  (
( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) 
<->  ( N  e.  NN  \/  -u N  e.  NN0 ) ) )
1211pm5.32i 449 . 2  |-  ( ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) )  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  -u N  e.  NN0 ) ) )
131, 12bitri 183 1  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  -u N  e.  NN0 ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    \/ wo 682    \/ w3o 946    = wceq 1316    e. wcel 1465   RRcr 7587   0cc0 7588   -ucneg 7902   NNcn 8684   NN0cn0 8935   ZZcz 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903  df-neg 7904  df-n0 8936  df-z 9013
This theorem is referenced by:  znnen  11822
  Copyright terms: Public domain W3C validator