Theorem bj-ccinftydisj 33100

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.

Step	Hyp	Ref	Expression
1		bj-inftyexpidisj 33097	. . . 4 ⊢ ¬ (inftyexpi ‘𝑦) ∈ ℂ
2	1	nex 1731	. . 3 ⊢ ¬ ∃𝑦(inftyexpi ‘𝑦) ∈ ℂ
3		elin 3796	. . . . . 6 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ∞))
4		df-bj-inftyexpi 33094	. . . . . . . . . . 11 ⊢ inftyexpi = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
5	4	funmpt2 5927	. . . . . . . . . 10 ⊢ Fun inftyexpi
6		elrnrexdm 6363	. . . . . . . . . 10 ⊢ (Fun inftyexpi → (𝑥 ∈ ran inftyexpi → ∃𝑦 ∈ dom inftyexpi 𝑥 = (inftyexpi ‘𝑦)))
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ (𝑥 ∈ ran inftyexpi → ∃𝑦 ∈ dom inftyexpi 𝑥 = (inftyexpi ‘𝑦))
8		rexex 3002	. . . . . . . . 9 ⊢ (∃𝑦 ∈ dom inftyexpi 𝑥 = (inftyexpi ‘𝑦) → ∃𝑦 𝑥 = (inftyexpi ‘𝑦))
9	7, 8	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ran inftyexpi → ∃𝑦 𝑥 = (inftyexpi ‘𝑦))
10		df-bj-ccinfty 33099	. . . . . . . 8 ⊢ ℂ∞ = ran inftyexpi
11	9, 10	eleq2s 2719	. . . . . . 7 ⊢ (𝑥 ∈ ℂ∞ → ∃𝑦 𝑥 = (inftyexpi ‘𝑦))
12	11	anim2i 593	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ∞) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)))
13	3, 12	sylbi 207	. . . . 5 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)))
14		ancom 466	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
15		exancom 1787	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) ↔ ∃𝑦(𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
16		19.41v 1914	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
17	15, 16	bitri 264	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
18	14, 17	sylbb2 228	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)) → ∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)))
19	13, 18	syl 17	. . . 4 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) → ∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)))
20		eleq1 2689	. . . . . 6 ⊢ (𝑥 = (inftyexpi ‘𝑦) → (𝑥 ∈ ℂ ↔ (inftyexpi ‘𝑦) ∈ ℂ))
21	20	biimpac 503	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) → (inftyexpi ‘𝑦) ∈ ℂ)
22	21	eximi 1762	. . . 4 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) → ∃𝑦(inftyexpi ‘𝑦) ∈ ℂ)
23	19, 22	syl 17	. . 3 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) → ∃𝑦(inftyexpi ‘𝑦) ∈ ℂ)
24	2, 23	mto 188	. 2 ⊢ ¬ 𝑥 ∈ (ℂ ∩ ℂ∞)
25	24	nel0 3932	1 ⊢ (ℂ ∩ ℂ∞) = ∅

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913   ∩ cin 3573  ∅c0 3915  ⟨cop 4183  dom cdm 5114  ran crn 5115  Fun wfun 5882  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  -cneg 10267  (,]cioc 12176  πcpi 14797  inftyexpi cinftyexpi 33093  ℂ_∞cccinfty 33098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-c 9942  df-bj-inftyexpi 33094  df-bj-ccinfty 33099

This theorem is referenced by: (None)