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Theorem bj-ccinftydisj 34668
 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 34665 . . . 4 ¬ (+∞ei𝑦) ∈ ℂ
21nex 1802 . . 3 ¬ ∃𝑦(+∞ei𝑦) ∈ ℂ
3 elin 3897 . . . . . 6 (𝑥 ∈ (ℂ ∩ ℂ) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ))
4 df-bj-inftyexpi 34662 . . . . . . . . . . 11 +∞ei = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
54funmpt2 6366 . . . . . . . . . 10 Fun +∞ei
6 elrnrexdm 6837 . . . . . . . . . 10 (Fun +∞ei → (𝑥 ∈ ran +∞ei → ∃𝑦 ∈ dom +∞ei𝑥 = (+∞ei𝑦)))
75, 6ax-mp 5 . . . . . . . . 9 (𝑥 ∈ ran +∞ei → ∃𝑦 ∈ dom +∞ei𝑥 = (+∞ei𝑦))
8 rexex 3203 . . . . . . . . 9 (∃𝑦 ∈ dom +∞ei𝑥 = (+∞ei𝑦) → ∃𝑦 𝑥 = (+∞ei𝑦))
97, 8syl 17 . . . . . . . 8 (𝑥 ∈ ran +∞ei → ∃𝑦 𝑥 = (+∞ei𝑦))
10 df-bj-ccinfty 34667 . . . . . . . 8 = ran +∞ei
119, 10eleq2s 2908 . . . . . . 7 (𝑥 ∈ ℂ → ∃𝑦 𝑥 = (+∞ei𝑦))
1211anim2i 619 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)))
133, 12sylbi 220 . . . . 5 (𝑥 ∈ (ℂ ∩ ℂ) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)))
14 ancom 464 . . . . . 6 ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei𝑦)) ↔ (∃𝑦 𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
15 exancom 1862 . . . . . . 7 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)) ↔ ∃𝑦(𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
16 19.41v 1950 . . . . . . 7 (∃𝑦(𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ) ↔ (∃𝑦 𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
1715, 16bitri 278 . . . . . 6 (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei𝑦)) ↔ (∃𝑦 𝑥 = (+∞ei𝑦) ∧ 𝑥 ∈ ℂ))
1814, 17sylbb2 241 . . . . 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 2877 . . . . . 6 (𝑥 = (+∞ei𝑦) → (𝑥 ∈ ℂ ↔ (+∞ei𝑦) ∈ ℂ))
2120biimpac 482 . . . . 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 200 . 2 ¬ 𝑥 ∈ (ℂ ∩ ℂ)
2524nel0 4264 1 (ℂ ∩ ℂ) = ∅
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107   ∩ cin 3880  ∅c0 4243  ⟨cop 4531  dom cdm 5520  ran crn 5521  Fun wfun 6321  ‘cfv 6327  (class class class)co 7140  ℂcc 10531  -cneg 10867  (,]cioc 12734  πcpi 15419  +∞eicinftyexpi 34661  ℂ∞cccinfty 34666 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7448  ax-reg 9047  ax-cnex 10589 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-iota 6286  df-fun 6329  df-fn 6330  df-fv 6335  df-c 10539  df-bj-inftyexpi 34662  df-bj-ccinfty 34667 This theorem is referenced by: (None)
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