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Theorem bj-ccinftydisj 33436
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 33433 . . . 4 ¬ (inftyexpi ‘𝑦) ∈ ℂ
21nex 1882 . . 3 ¬ ∃𝑦(inftyexpi ‘𝑦) ∈ ℂ
3 elin 4006 . . . . . 6 (𝑥 ∈ (ℂ ∩ ℂ) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ))
4 df-bj-inftyexpi 33430 . . . . . . . . . . 11 inftyexpi = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
54funmpt2 6150 . . . . . . . . . 10 Fun inftyexpi
6 elrnrexdm 6595 . . . . . . . . . 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 3200 . . . . . . . . 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 33435 . . . . . . . 8 = ran inftyexpi
119, 10eleq2s 2914 . . . . . . 7 (𝑥 ∈ ℂ → ∃𝑦 𝑥 = (inftyexpi ‘𝑦))
1211anim2i 605 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)))
133, 12sylbi 208 . . . . 5 (𝑥 ∈ (ℂ ∩ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)))
14 ancom 450 . . . . . 6 ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
15 exancom 1947 . . . . . . 7 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) ↔ ∃𝑦(𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
16 19.41v 2040 . . . . . . 7 (∃𝑦(𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
1715, 16bitri 266 . . . . . 6 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
1814, 17sylbb2 229 . . . . 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 2884 . . . . . 6 (𝑥 = (inftyexpi ‘𝑦) → (𝑥 ∈ ℂ ↔ (inftyexpi ‘𝑦) ∈ ℂ))
2120biimpac 466 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) → (inftyexpi ‘𝑦) ∈ ℂ)
2221eximi 1919 . . . 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 4144 1 (ℂ ∩ ℂ) = ∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wex 1859  wcel 2157  wrex 3108  cin 3779  c0 4127  cop 4387  dom cdm 5324  ran crn 5325  Fun wfun 6105  cfv 6111  (class class class)co 6884  cc 10229  -cneg 10562  (,]cioc 12414  πcpi 15037  inftyexpi cinftyexpi 33429  cccinfty 33434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189  ax-reg 8746  ax-cnex 10287
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-iota 6074  df-fun 6113  df-fn 6114  df-fv 6119  df-c 10237  df-bj-inftyexpi 33430  df-bj-ccinfty 33435
This theorem is referenced by: (None)
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