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Metamath Proof Explorer

Theorem 0cnALT2

Description: Alternate proof of 0cnALT which is shorter, but depends on ax-8 , ax-13 , ax-sep , ax-nul , ax-pow , ax-pr , ax-un , and every complex number axiom except ax-pre-mulgt0 and ax-pre-sup . (Contributed by NM, 19-Feb-2005) (Revised by Mario Carneiro, 27-May-2016) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 0cnALT2 
|- 0 e. CC

Proof

Step Hyp Ref Expression
1 ax-icn 
 |-  _i e. CC
2 cnegex 
 |-  ( _i e. CC -> E. x e. CC ( _i + x ) = 0 )
3 1 2 ax-mp 
 |-  E. x e. CC ( _i + x ) = 0
4 addcl 
 |-  ( ( _i e. CC /\ x e. CC ) -> ( _i + x ) e. CC )
5 1 4 mpan 
 |-  ( x e. CC -> ( _i + x ) e. CC )
6 eleq1 
 |-  ( ( _i + x ) = 0 -> ( ( _i + x ) e. CC <-> 0 e. CC ) )
7 5 6 syl5ibcom 
 |-  ( x e. CC -> ( ( _i + x ) = 0 -> 0 e. CC ) )
8 7 rexlimiv 
 |-  ( E. x e. CC ( _i + x ) = 0 -> 0 e. CC )
9 3 8 ax-mp 
 |-  0 e. CC