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Theorem 0cnALT 7651
Description: Alternate proof of 0cn 7459. (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 ∈ ℂ

Proof of Theorem 0cnALT
StepHypRef Expression
1 ax-icn 7419 . . 3 i ∈ ℂ
2 cnegex 7639 . . 3 (i ∈ ℂ → ∃𝑥 ∈ ℂ (i + 𝑥) = 0)
31, 2ax-mp 7 . 2 𝑥 ∈ ℂ (i + 𝑥) = 0
4 addcl 7446 . . . . 5 ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i + 𝑥) ∈ ℂ)
51, 4mpan 415 . . . 4 (𝑥 ∈ ℂ → (i + 𝑥) ∈ ℂ)
6 eleq1 2150 . . . 4 ((i + 𝑥) = 0 → ((i + 𝑥) ∈ ℂ ↔ 0 ∈ ℂ))
75, 6syl5ibcom 153 . . 3 (𝑥 ∈ ℂ → ((i + 𝑥) = 0 → 0 ∈ ℂ))
87rexlimiv 2483 . 2 (∃𝑥 ∈ ℂ (i + 𝑥) = 0 → 0 ∈ ℂ)
93, 8ax-mp 7 1 0 ∈ ℂ
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  wrex 2360  (class class class)co 5634  cc 7327  0cc0 7329  ici 7331   + caddc 7332
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 7416  ax-1cn 7417  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435
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 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637
This theorem is referenced by: (None)
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