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Theorem bj-ccinftydisj 36398
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 36395 . . . 4 ¬ (+∞ei𝑦) ∈ ℂ
21nex 1801 . . 3 ¬ ∃𝑦(+∞ei𝑦) ∈ ℂ
3 elin 3965 . . . . . 6 (𝑥 ∈ (ℂ ∩ ℂ) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ))
4 df-bj-inftyexpi 36392 . . . . . . . . . . 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 3075 . . . . . . . . 9 (∃𝑦 ∈ dom +∞ei𝑥 = (+∞ei𝑦) → ∃𝑦 𝑥 = (+∞ei𝑦))
97, 8syl 17 . . . . . . . 8 (𝑥 ∈ ran +∞ei → ∃𝑦 𝑥 = (+∞ei𝑦))
10 df-bj-ccinfty 36397 . . . . . . . 8 = ran +∞ei
119, 10eleq2s 2850 . . . . . . 7 (𝑥 ∈ ℂ → ∃𝑦 𝑥 = (+∞ei𝑦))
1211anim2i 616 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)))
133, 12sylbi 216 . . . . 5 (𝑥 ∈ (ℂ ∩ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)))
14 ancom 460 . . . . . 6 ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)) ↔ (∃𝑦 𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
15 exancom 1863 . . . . . . 7 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)) ↔ ∃𝑦(𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
16 19.41v 1952 . . . . . . 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 2820 . . . . . 6 (𝑥 = (+∞ei𝑦) → (𝑥 ∈ ℂ ↔ (+∞ei𝑦) ∈ ℂ))
2120biimpac 478 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)) → (+∞ei𝑦) ∈ ℂ)
2221eximi 1836 . . . 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 395   = wceq 1540  wex 1780  wcel 2105  wrex 3069  cin 3948  c0 4323  cop 4635  dom cdm 5677  ran crn 5678  Fun wfun 6538  cfv 6544  (class class class)co 7412  cc 11111  -cneg 11450  (,]cioc 13330  πcpi 16015  +∞eicinftyexpi 36391  cccinfty 36396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728  ax-reg 9590  ax-cnex 11169
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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 11119  df-bj-inftyexpi 36392  df-bj-ccinfty 36397
This theorem is referenced by: (None)
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