Theorem bj-ccinftydisj 35311

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 35308	. . . 4 ⊢ ¬ (+∞ei‘𝑦) ∈ ℂ
2	1	nex 1804	. . 3 ⊢ ¬ ∃𝑦(+∞ei‘𝑦) ∈ ℂ
3		elin 3899	. . . . . 6 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ∞))
4		df-bj-inftyexpi 35305	. . . . . . . . . . 11 ⊢ +∞ei = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
5	4	funmpt2 6457	. . . . . . . . . 10 ⊢ Fun +∞ei
6		elrnrexdm 6947	. . . . . . . . . 10 ⊢ (Fun +∞ei → (𝑥 ∈ ran +∞ei → ∃𝑦 ∈ dom +∞ei𝑥 = (+∞ei‘𝑦)))
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ (𝑥 ∈ ran +∞ei → ∃𝑦 ∈ dom +∞ei𝑥 = (+∞ei‘𝑦))
8		rexex 3167	. . . . . . . . 9 ⊢ (∃𝑦 ∈ dom +∞ei𝑥 = (+∞ei‘𝑦) → ∃𝑦 𝑥 = (+∞ei‘𝑦))
9	7, 8	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ran +∞ei → ∃𝑦 𝑥 = (+∞ei‘𝑦))
10		df-bj-ccinfty 35310	. . . . . . . 8 ⊢ ℂ∞ = ran +∞ei
11	9, 10	eleq2s 2857	. . . . . . 7 ⊢ (𝑥 ∈ ℂ∞ → ∃𝑦 𝑥 = (+∞ei‘𝑦))
12	11	anim2i 616	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ∞) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei‘𝑦)))
13	3, 12	sylbi 216	. . . . 5 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei‘𝑦)))
14		ancom 460	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei‘𝑦)) ↔ (∃𝑦 𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
15		exancom 1865	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)) ↔ ∃𝑦(𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
16		19.41v 1954	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ) ↔ (∃𝑦 𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
17	15, 16	bitri 274	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)) ↔ (∃𝑦 𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
18	14, 17	sylbb2 237	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei‘𝑦)) → ∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)))
19	13, 18	syl 17	. . . 4 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) → ∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)))
20		eleq1 2826	. . . . . 6 ⊢ (𝑥 = (+∞ei‘𝑦) → (𝑥 ∈ ℂ ↔ (+∞ei‘𝑦) ∈ ℂ))
21	20	biimpac 478	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)) → (+∞ei‘𝑦) ∈ ℂ)
22	21	eximi 1838	. . . 4 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)) → ∃𝑦(+∞ei‘𝑦) ∈ ℂ)
23	19, 22	syl 17	. . 3 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) → ∃𝑦(+∞ei‘𝑦) ∈ ℂ)
24	2, 23	mto 196	. 2 ⊢ ¬ 𝑥 ∈ (ℂ ∩ ℂ∞)
25	24	nel0 4281	1 ⊢ (ℂ ∩ ℂ∞) = ∅

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃wrex 3064   ∩ cin 3882  ∅c0 4253  ⟨cop 4564  dom cdm 5580  ran crn 5581  Fun wfun 6412  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  -cneg 11136  (,]cioc 13009  πcpi 15704  +∞eⁱcinftyexpi 35304  ℂ_∞cccinfty 35309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-cnex 10858

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-c 10808  df-bj-inftyexpi 35305  df-bj-ccinfty 35310

This theorem is referenced by: (None)