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Theorem bj-ccinftydisj 33230
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 33227 . . . 4 ¬ (inftyexpi ‘𝑦) ∈ ℂ
21nex 1771 . . 3 ¬ ∃𝑦(inftyexpi ‘𝑦) ∈ ℂ
3 elin 3829 . . . . . 6 (𝑥 ∈ (ℂ ∩ ℂ) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ))
4 df-bj-inftyexpi 33224 . . . . . . . . . . 11 inftyexpi = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
54funmpt2 5965 . . . . . . . . . 10 Fun inftyexpi
6 elrnrexdm 6403 . . . . . . . . . 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 3031 . . . . . . . . 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 33229 . . . . . . . 8 = ran inftyexpi
119, 10eleq2s 2748 . . . . . . 7 (𝑥 ∈ ℂ → ∃𝑦 𝑥 = (inftyexpi ‘𝑦))
1211anim2i 592 . . . . . 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 1827 . . . . . . 7 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) ↔ ∃𝑦(𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
16 19.41v 1917 . . . . . . 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 2718 . . . . . 6 (𝑥 = (inftyexpi ‘𝑦) → (𝑥 ∈ ℂ ↔ (inftyexpi ‘𝑦) ∈ ℂ))
2120biimpac 502 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) → (inftyexpi ‘𝑦) ∈ ℂ)
2221eximi 1802 . . . 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 3965 1 (ℂ ∩ ℂ) = ∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  cin 3606  c0 3948  cop 4216  dom cdm 5143  ran crn 5144  Fun wfun 5920  cfv 5926  (class class class)co 6690  cc 9972  -cneg 10305  (,]cioc 12214  πcpi 14841  inftyexpi cinftyexpi 33223  cccinfty 33228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-cnex 10030
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-c 9980  df-bj-inftyexpi 33224  df-bj-ccinfty 33229
This theorem is referenced by: (None)
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