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Theorem bj-ccinftydisj 33071
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 33068 . . . 4 ¬ (inftyexpi ‘𝑦) ∈ ℂ
21nex 1729 . . 3 ¬ ∃𝑦(inftyexpi ‘𝑦) ∈ ℂ
3 elin 3788 . . . . . 6 (𝑥 ∈ (ℂ ∩ ℂ) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ))
4 df-bj-inftyexpi 33065 . . . . . . . . . . 11 inftyexpi = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
54funmpt2 5915 . . . . . . . . . 10 Fun inftyexpi
6 elrnrexdm 6349 . . . . . . . . . 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 2999 . . . . . . . . 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 33070 . . . . . . . 8 = ran inftyexpi
119, 10eleq2s 2717 . . . . . . 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 466 . . . . . 6 ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
15 exancom 1785 . . . . . . 7 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) ↔ ∃𝑦(𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
16 19.41v 1912 . . . . . . 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 2687 . . . . . 6 (𝑥 = (inftyexpi ‘𝑦) → (𝑥 ∈ ℂ ↔ (inftyexpi ‘𝑦) ∈ ℂ))
2120biimpac 503 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) → (inftyexpi ‘𝑦) ∈ ℂ)
2221eximi 1760 . . . 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 3924 1 (ℂ ∩ ℂ) = ∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  wrex 2910  cin 3566  c0 3907  cop 4174  dom cdm 5104  ran crn 5105  Fun wfun 5870  cfv 5876  (class class class)co 6635  cc 9919  -cneg 10252  (,]cioc 12161  πcpi 14778  inftyexpi cinftyexpi 33064  cccinfty 33069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-reg 8482  ax-cnex 9977
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884  df-c 9927  df-bj-inftyexpi 33065  df-bj-ccinfty 33070
This theorem is referenced by: (None)
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