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Theorem 0cnALT 8143
Description: Alternate proof of 0cn 7946. (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 7903 . . 3  |-  _i  e.  CC
2 cnegex 8131 . . 3  |-  ( _i  e.  CC  ->  E. x  e.  CC  ( _i  +  x )  =  0 )
31, 2ax-mp 5 . 2  |-  E. x  e.  CC  ( _i  +  x )  =  0
4 addcl 7933 . . . . 5  |-  ( ( _i  e.  CC  /\  x  e.  CC )  ->  ( _i  +  x
)  e.  CC )
51, 4mpan 424 . . . 4  |-  ( x  e.  CC  ->  (
_i  +  x )  e.  CC )
6 eleq1 2240 . . . 4  |-  ( ( _i  +  x )  =  0  ->  (
( _i  +  x
)  e.  CC  <->  0  e.  CC ) )
75, 6syl5ibcom 155 . . 3  |-  ( x  e.  CC  ->  (
( _i  +  x
)  =  0  -> 
0  e.  CC ) )
87rexlimiv 2588 . 2  |-  ( E. x  e.  CC  (
_i  +  x )  =  0  ->  0  e.  CC )
93, 8ax-mp 5 1  |-  0  e.  CC
Colors of variables: wff set class
Syntax hints:    = wceq 1353    e. wcel 2148   E.wrex 2456  (class class class)co 5872   CCcc 7806   0cc0 7808   _ici 7810    + caddc 7811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-resscn 7900  ax-1cn 7901  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-distr 7912  ax-i2m1 7913  ax-0id 7916  ax-rnegex 7917  ax-cnre 7919
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fv 5223  df-ov 5875
This theorem is referenced by: (None)
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