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Theorem 0cnALT 8209
Description: Alternate proof of 0cn 8011. (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 7967 . . 3 i ∈ ℂ
2 cnegex 8197 . . 3 (i ∈ ℂ → ∃𝑥 ∈ ℂ (i + 𝑥) = 0)
31, 2ax-mp 5 . 2 𝑥 ∈ ℂ (i + 𝑥) = 0
4 addcl 7997 . . . . 5 ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i + 𝑥) ∈ ℂ)
51, 4mpan 424 . . . 4 (𝑥 ∈ ℂ → (i + 𝑥) ∈ ℂ)
6 eleq1 2256 . . . 4 ((i + 𝑥) = 0 → ((i + 𝑥) ∈ ℂ ↔ 0 ∈ ℂ))
75, 6syl5ibcom 155 . . 3 (𝑥 ∈ ℂ → ((i + 𝑥) = 0 → 0 ∈ ℂ))
87rexlimiv 2605 . 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 2164  wrex 2473  (class class class)co 5918  cc 7870  0cc0 7872  ici 7874   + caddc 7875
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 2175  ax-resscn 7964  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921
This theorem is referenced by: (None)
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