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Theorem bj-ccinftydisj 36094
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 36091 . . . 4 ¬ (+∞ei𝑦) ∈ ℂ
21nex 1803 . . 3 ¬ ∃𝑦(+∞ei𝑦) ∈ ℂ
3 elin 3965 . . . . . 6 (𝑥 ∈ (ℂ ∩ ℂ) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ))
4 df-bj-inftyexpi 36088 . . . . . . . . . . 11 +∞ei = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
54funmpt2 6588 . . . . . . . . . 10 Fun +∞ei
6 elrnrexdm 7091 . . . . . . . . . 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 36093 . . . . . . . 8 = ran +∞ei
119, 10eleq2s 2852 . . . . . . 7 (𝑥 ∈ ℂ → ∃𝑦 𝑥 = (+∞ei𝑦))
1211anim2i 618 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)))
133, 12sylbi 216 . . . . 5 (𝑥 ∈ (ℂ ∩ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)))
14 ancom 462 . . . . . 6 ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)) ↔ (∃𝑦 𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
15 exancom 1865 . . . . . . 7 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)) ↔ ∃𝑦(𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
16 19.41v 1954 . . . . . . 7 (∃𝑦(𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ) ↔ (∃𝑦 𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
1715, 16bitri 275 . . . . . 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 2822 . . . . . 6 (𝑥 = (+∞ei𝑦) → (𝑥 ∈ ℂ ↔ (+∞ei𝑦) ∈ ℂ))
2120biimpac 480 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)) → (+∞ei𝑦) ∈ ℂ)
2221eximi 1838 . . . 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 4351 1 (ℂ ∩ ℂ) = ∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wrex 3071  cin 3948  c0 4323  cop 4635  dom cdm 5677  ran crn 5678  Fun wfun 6538  cfv 6544  (class class class)co 7409  cc 11108  -cneg 11445  (,]cioc 13325  πcpi 16010  +∞eicinftyexpi 36087  cccinfty 36092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-cnex 11166
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-c 11116  df-bj-inftyexpi 36088  df-bj-ccinfty 36093
This theorem is referenced by: (None)
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