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Theorem bj-ccinftydisj 35616
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 35613 . . . 4 ¬ (+∞ei𝑦) ∈ ℂ
21nex 1802 . . 3 ¬ ∃𝑦(+∞ei𝑦) ∈ ℂ
3 elin 3924 . . . . . 6 (𝑥 ∈ (ℂ ∩ ℂ) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ))
4 df-bj-inftyexpi 35610 . . . . . . . . . . 11 +∞ei = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
54funmpt2 6537 . . . . . . . . . 10 Fun +∞ei
6 elrnrexdm 7035 . . . . . . . . . 10 (Fun +∞ei → (𝑥 ∈ ran +∞ei → ∃𝑦 ∈ dom +∞ei𝑥 = (+∞ei𝑦)))
75, 6ax-mp 5 . . . . . . . . 9 (𝑥 ∈ ran +∞ei → ∃𝑦 ∈ dom +∞ei𝑥 = (+∞ei𝑦))
8 rexex 3077 . . . . . . . . 9 (∃𝑦 ∈ dom +∞ei𝑥 = (+∞ei𝑦) → ∃𝑦 𝑥 = (+∞ei𝑦))
97, 8syl 17 . . . . . . . 8 (𝑥 ∈ ran +∞ei → ∃𝑦 𝑥 = (+∞ei𝑦))
10 df-bj-ccinfty 35615 . . . . . . . 8 = ran +∞ei
119, 10eleq2s 2856 . . . . . . 7 (𝑥 ∈ ℂ → ∃𝑦 𝑥 = (+∞ei𝑦))
1211anim2i 617 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)))
133, 12sylbi 216 . . . . 5 (𝑥 ∈ (ℂ ∩ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)))
14 ancom 461 . . . . . 6 ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)) ↔ (∃𝑦 𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
15 exancom 1864 . . . . . . 7 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)) ↔ ∃𝑦(𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
16 19.41v 1953 . . . . . . 7 (∃𝑦(𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ) ↔ (∃𝑦 𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
1715, 16bitri 274 . . . . . 6 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)) ↔ (∃𝑦 𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
1814, 17sylbb2 237 . . . . 5 ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)) → ∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)))
1913, 18syl 17 . . . 4 (𝑥 ∈ (ℂ ∩ ℂ) → ∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)))
20 eleq1 2825 . . . . . 6 (𝑥 = (+∞ei𝑦) → (𝑥 ∈ ℂ ↔ (+∞ei𝑦) ∈ ℂ))
2120biimpac 479 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)) → (+∞ei𝑦) ∈ ℂ)
2221eximi 1837 . . . 4 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)) → ∃𝑦(+∞ei𝑦) ∈ ℂ)
2319, 22syl 17 . . 3 (𝑥 ∈ (ℂ ∩ ℂ) → ∃𝑦(+∞ei𝑦) ∈ ℂ)
242, 23mto 196 . 2 ¬ 𝑥 ∈ (ℂ ∩ ℂ)
2524nel0 4308 1 (ℂ ∩ ℂ) = ∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  wrex 3071  cin 3907  c0 4280  cop 4590  dom cdm 5631  ran crn 5632  Fun wfun 6487  cfv 6493  (class class class)co 7351  cc 11007  -cneg 11344  (,]cioc 13219  πcpi 15903  +∞eicinftyexpi 35609  cccinfty 35614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664  ax-reg 9486  ax-cnex 11065
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6445  df-fun 6495  df-fn 6496  df-fv 6501  df-c 11015  df-bj-inftyexpi 35610  df-bj-ccinfty 35615
This theorem is referenced by: (None)
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