Theorem 0cnALT 8479

Description: Alternate proof of 0cn 8282. (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

Step	Hyp	Ref	Expression
1		ax-icn 8238	. . 3 |- _i e. CC
2		cnegex 8467	. . 3 |- ( _i e. CC -> E. x e. CC ( _i + x ) = 0 )
3	1, 2	ax-mp 5	. 2 |- E. x e. CC ( _i + x ) = 0
4		addcl 8268	. . . . 5 |- ( ( _i e. CC /\ x e. CC ) -> ( _i + x ) e. CC )
5	1, 4	mpan 424	. . . 4 |- ( x e. CC -> ( _i + x ) e. CC )
6		eleq1 2297	. . . 4 |- ( ( _i + x ) = 0 -> ( ( _i + x ) e. CC <-> 0 e. CC ) )
7	5, 6	syl5ibcom 155	. . 3 |- ( x e. CC -> ( ( _i + x ) = 0 -> 0 e. CC ) )
8	7	rexlimiv 2656	. 2 |- ( E. x e. CC ( _i + x ) = 0 -> 0 e. CC )
9	3, 8	ax-mp 5	1 |- 0 e. CC

Syntax hints:   = wceq 1398   e. wcel 2205   E.wrex 2523  (class class class)co 6058   CCcc 8141   0cc0 8143   _ici 8145   + caddc 8146

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-resscn 8235  ax-1cn 8236  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061

This theorem is referenced by: (None)