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Theorem elzdif0 28796
Description: Lemma for qqhval2 28798. (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 3557 . . 3  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  M  e.  ZZ )
2 eldifn 3558 . . 3  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  -.  M  e.  { 0 } )
3 elsncg 3993 . . . . 5  |-  ( M  e.  ZZ  ->  ( M  e.  { 0 } 
<->  M  =  0 ) )
43notbid 296 . . . 4  |-  ( M  e.  ZZ  ->  ( -.  M  e.  { 0 }  <->  -.  M  = 
0 ) )
54biimpa 487 . . 3  |-  ( ( M  e.  ZZ  /\  -.  M  e.  { 0 } )  ->  -.  M  =  0 )
61, 2, 5syl2anc 667 . 2  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  -.  M  = 
0 )
7 elz 10946 . . . . 5  |-  ( M  e.  ZZ  <->  ( M  e.  RR  /\  ( M  =  0  \/  M  e.  NN  \/  -u M  e.  NN ) ) )
81, 7sylib 200 . . . 4  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  ( M  e.  RR  /\  ( M  =  0  \/  M  e.  NN  \/  -u M  e.  NN ) ) )
98simprd 465 . . 3  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  ( M  =  0  \/  M  e.  NN  \/  -u M  e.  NN ) )
10 3orass 989 . . 3  |-  ( ( M  =  0  \/  M  e.  NN  \/  -u M  e.  NN )  <-> 
( M  =  0  \/  ( M  e.  NN  \/  -u M  e.  NN ) ) )
119, 10sylib 200 . 2  |-  ( M  e.  ( ZZ  \  { 0 } )  ->  ( M  =  0  \/  ( M  e.  NN  \/  -u M  e.  NN ) ) )
12 orel1 384 . 2  |-  ( -.  M  =  0  -> 
( ( M  =  0  \/  ( M  e.  NN  \/  -u M  e.  NN ) )  -> 
( M  e.  NN  \/  -u M  e.  NN ) ) )
136, 11, 12sylc 62 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 370    /\ wa 371    \/ w3o 985    = wceq 1446    e. wcel 1889    \ cdif 3403   {csn 3970   RRcr 9543   0cc0 9544   -ucneg 9866   NNcn 10616   ZZcz 10944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-iota 5549  df-fv 5593  df-ov 6298  df-neg 9868  df-z 10945
This theorem is referenced by: (None)
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