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Theorem elzdif0 28209
Description: Lemma for qqhval2 28211 (Contributed by Thierry Arnoux, 29-Oct-2017.)
Assertion
Ref Expression
elzdif0  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  ( M  e.  NN  \/  -u M  e.  NN ) )

Proof of Theorem elzdif0
StepHypRef Expression
1 eldifi 3622 . . 3  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  M  e.  ZZ )
2 eldifn 3623 . . 3  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  -.  M  e.  { 0 } )
3 elsncg 4055 . . . . 5  |-  ( M  e.  ZZ  ->  ( M  e.  { 0 } 
<->  M  =  0 ) )
43notbid 294 . . . 4  |-  ( M  e.  ZZ  ->  ( -.  M  e.  { 0 }  <->  -.  M  = 
0 ) )
54biimpa 484 . . 3  |-  ( ( M  e.  ZZ  /\  -.  M  e.  { 0 } )  ->  -.  M  =  0 )
61, 2, 5syl2anc 661 . 2  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  -.  M  = 
0 )
7 elz 10887 . . . . 5  |-  ( M  e.  ZZ  <->  ( M  e.  RR  /\  ( M  =  0  \/  M  e.  NN  \/  -u M  e.  NN ) ) )
81, 7sylib 196 . . . 4  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  ( M  e.  RR  /\  ( M  =  0  \/  M  e.  NN  \/  -u M  e.  NN ) ) )
98simprd 463 . . 3  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  ( M  =  0  \/  M  e.  NN  \/  -u M  e.  NN ) )
10 3orass 976 . . 3  |-  ( ( M  =  0  \/  M  e.  NN  \/  -u M  e.  NN )  <-> 
( M  =  0  \/  ( M  e.  NN  \/  -u M  e.  NN ) ) )
119, 10sylib 196 . 2  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  ( M  =  0  \/  ( M  e.  NN  \/  -u M  e.  NN ) ) )
12 orel1 382 . 2  |-  ( -.  M  =  0  -> 
( ( M  =  0  \/  ( M  e.  NN  \/  -u M  e.  NN ) )  -> 
( M  e.  NN  \/  -u M  e.  NN ) ) )
136, 11, 12sylc 60 1  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  ( M  e.  NN  \/  -u M  e.  NN ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 972    = wceq 1395    e. wcel 1819    \ cdif 3468   {csn 4032   RRcr 9508   0cc0 9509   -ucneg 9825   NNcn 10556   ZZcz 10885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-neg 9827  df-z 10886
This theorem is referenced by: (None)
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