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Theorem 0cnALT 7662
Description: Alternate proof of 0cn 7470. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
0cnALT  |-  0  e.  CC

Proof of Theorem 0cnALT
StepHypRef Expression
1 ax-icn 7430 . . 3  |-  _i  e.  CC
2 cnegex 7650 . . 3  |-  ( _i  e.  CC  ->  E. x  e.  CC  ( _i  +  x )  =  0 )
31, 2ax-mp 7 . 2  |-  E. x  e.  CC  ( _i  +  x )  =  0
4 addcl 7457 . . . . 5  |-  ( ( _i  e.  CC  /\  x  e.  CC )  ->  ( _i  +  x
)  e.  CC )
51, 4mpan 415 . . . 4  |-  ( x  e.  CC  ->  (
_i  +  x )  e.  CC )
6 eleq1 2150 . . . 4  |-  ( ( _i  +  x )  =  0  ->  (
( _i  +  x
)  e.  CC  <->  0  e.  CC ) )
75, 6syl5ibcom 153 . . 3  |-  ( x  e.  CC  ->  (
( _i  +  x
)  =  0  -> 
0  e.  CC ) )
87rexlimiv 2483 . 2  |-  ( E. x  e.  CC  (
_i  +  x )  =  0  ->  0  e.  CC )
93, 8ax-mp 7 1  |-  0  e.  CC
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   E.wrex 2360  (class class class)co 5644   CCcc 7338   0cc0 7340   _ici 7342    + caddc 7343
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-resscn 7427  ax-1cn 7428  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-addcom 7435  ax-addass 7437  ax-distr 7439  ax-i2m1 7440  ax-0id 7443  ax-rnegex 7444  ax-cnre 7446
This theorem depends on definitions:  df-bi 115  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-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-iota 4975  df-fv 5018  df-ov 5647
This theorem is referenced by: (None)
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