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Theorem bj-ccinftydisj 33382
Description: The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-ccinftydisj (ℂ ∩ ℂ) = ∅

Proof of Theorem bj-ccinftydisj
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-inftyexpidisj 33379 . . . 4 ¬ (inftyexpi ‘𝑦) ∈ ℂ
21nex 1868 . . 3 ¬ ∃𝑦(inftyexpi ‘𝑦) ∈ ℂ
3 elin 3927 . . . . . 6 (𝑥 ∈ (ℂ ∩ ℂ) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ))
4 df-bj-inftyexpi 33376 . . . . . . . . . . 11 inftyexpi = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
54funmpt2 6076 . . . . . . . . . 10 Fun inftyexpi
6 elrnrexdm 6514 . . . . . . . . . 10 (Fun inftyexpi → (𝑥 ∈ ran inftyexpi → ∃𝑦 ∈ dom inftyexpi 𝑥 = (inftyexpi ‘𝑦)))
75, 6ax-mp 5 . . . . . . . . 9 (𝑥 ∈ ran inftyexpi → ∃𝑦 ∈ dom inftyexpi 𝑥 = (inftyexpi ‘𝑦))
8 rexex 3128 . . . . . . . . 9 (∃𝑦 ∈ dom inftyexpi 𝑥 = (inftyexpi ‘𝑦) → ∃𝑦 𝑥 = (inftyexpi ‘𝑦))
97, 8syl 17 . . . . . . . 8 (𝑥 ∈ ran inftyexpi → ∃𝑦 𝑥 = (inftyexpi ‘𝑦))
10 df-bj-ccinfty 33381 . . . . . . . 8 = ran inftyexpi
119, 10eleq2s 2845 . . . . . . 7 (𝑥 ∈ ℂ → ∃𝑦 𝑥 = (inftyexpi ‘𝑦))
1211anim2i 594 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)))
133, 12sylbi 207 . . . . 5 (𝑥 ∈ (ℂ ∩ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)))
14 ancom 465 . . . . . 6 ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
15 exancom 1924 . . . . . . 7 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) ↔ ∃𝑦(𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
16 19.41v 2014 . . . . . . 7 (∃𝑦(𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
1715, 16bitri 264 . . . . . 6 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
1814, 17sylbb2 228 . . . . 5 ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)) → ∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)))
1913, 18syl 17 . . . 4 (𝑥 ∈ (ℂ ∩ ℂ) → ∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)))
20 eleq1 2815 . . . . . 6 (𝑥 = (inftyexpi ‘𝑦) → (𝑥 ∈ ℂ ↔ (inftyexpi ‘𝑦) ∈ ℂ))
2120biimpac 504 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) → (inftyexpi ‘𝑦) ∈ ℂ)
2221eximi 1899 . . . 4 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) → ∃𝑦(inftyexpi ‘𝑦) ∈ ℂ)
2319, 22syl 17 . . 3 (𝑥 ∈ (ℂ ∩ ℂ) → ∃𝑦(inftyexpi ‘𝑦) ∈ ℂ)
242, 23mto 188 . 2 ¬ 𝑥 ∈ (ℂ ∩ ℂ)
2524nel0 4063 1 (ℂ ∩ ℂ) = ∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wex 1841  wcel 2127  wrex 3039  cin 3702  c0 4046  cop 4315  dom cdm 5254  ran crn 5255  Fun wfun 6031  cfv 6037  (class class class)co 6801  cc 10097  -cneg 10430  (,]cioc 12340  πcpi 14967  inftyexpi cinftyexpi 33375  cccinfty 33380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-reg 8650  ax-cnex 10155
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fn 6040  df-fv 6045  df-c 10105  df-bj-inftyexpi 33376  df-bj-ccinfty 33381
This theorem is referenced by: (None)
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