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Theorem 0cnALT 8195
Description: Alternate proof of 0cn 7997. (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 7953 . . 3 i ∈ ℂ
2 cnegex 8183 . . 3 (i ∈ ℂ → ∃𝑥 ∈ ℂ (i + 𝑥) = 0)
31, 2ax-mp 5 . 2 𝑥 ∈ ℂ (i + 𝑥) = 0
4 addcl 7983 . . . . 5 ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i + 𝑥) ∈ ℂ)
51, 4mpan 424 . . . 4 (𝑥 ∈ ℂ → (i + 𝑥) ∈ ℂ)
6 eleq1 2252 . . . 4 ((i + 𝑥) = 0 → ((i + 𝑥) ∈ ℂ ↔ 0 ∈ ℂ))
75, 6syl5ibcom 155 . . 3 (𝑥 ∈ ℂ → ((i + 𝑥) = 0 → 0 ∈ ℂ))
87rexlimiv 2601 . 2 (∃𝑥 ∈ ℂ (i + 𝑥) = 0 → 0 ∈ ℂ)
93, 8ax-mp 5 1 0 ∈ ℂ
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wcel 2160  wrex 2469  (class class class)co 5906  cc 7856  0cc0 7858  ici 7860   + caddc 7861
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-resscn 7950  ax-1cn 7951  ax-icn 7953  ax-addcl 7954  ax-addrcl 7955  ax-mulcl 7956  ax-addcom 7958  ax-addass 7960  ax-distr 7962  ax-i2m1 7963  ax-0id 7966  ax-rnegex 7967  ax-cnre 7969
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2758  df-un 3153  df-in 3155  df-ss 3162  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-br 4026  df-iota 5203  df-fv 5250  df-ov 5909
This theorem is referenced by: (None)
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