Theorem bj-ccinftydisj 32277

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 32274	. . . 4 ⊢ ¬ (inftyexpi ‘𝑦) ∈ ℂ
2	1	nex 1722	. . 3 ⊢ ¬ ∃𝑦(inftyexpi ‘𝑦) ∈ ℂ
3		elin 3758	. . . . . 6 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ∞))
4		df-bj-inftyexpi 32271	. . . . . . . . . . 11 ⊢ inftyexpi = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
5	4	funmpt2 5841	. . . . . . . . . 10 ⊢ Fun inftyexpi
6		elrnrexdm 6271	. . . . . . . . . 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 2985	. . . . . . . . 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 32276	. . . . . . . 8 ⊢ ℂ∞ = ran inftyexpi
11	9, 10	eleq2s 2706	. . . . . . 7 ⊢ (𝑥 ∈ ℂ∞ → ∃𝑦 𝑥 = (inftyexpi ‘𝑦))
12	11	anim2i 591	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ∞) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)))
13	3, 12	sylbi 206	. . . . 5 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)))
14		ancom 465	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (inftyexpi ‘𝑦)) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
15		exancom 1774	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) ↔ ∃𝑦(𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
16		19.41v 1901	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
17	15, 16	bitri 263	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) ↔ (∃𝑦 𝑥 = (inftyexpi ‘𝑦) ∧ 𝑥 ∈ ℂ))
18	14, 17	sylbb2 227	. . . . 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 2676	. . . . . 6 ⊢ (𝑥 = (inftyexpi ‘𝑦) → (𝑥 ∈ ℂ ↔ (inftyexpi ‘𝑦) ∈ ℂ))
21	20	biimpac 502	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 = (inftyexpi ‘𝑦)) → (inftyexpi ‘𝑦) ∈ ℂ)
22	21	eximi 1752	. . . 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 187	. 2 ⊢ ¬ 𝑥 ∈ (ℂ ∩ ℂ∞)
25	24	bj-nel0 32128	1 ⊢ (ℂ ∩ ℂ∞) = ∅

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897   ∩ cin 3539  ∅c0 3874  ⟨cop 4131  dom cdm 5038  ran crn 5039  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  -cneg 10146  (,]cioc 12047  πcpi 14636  inftyexpi cinftyexpi 32270  ℂ_∞cccinfty 32275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-cnex 9871

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-c 9821  df-bj-inftyexpi 32271  df-bj-ccinfty 32276

This theorem is referenced by: (None)