Theorem bj-ccinftydisj 37214

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 37211	. . . 4 ⊢ ¬ (+∞ei‘𝑦) ∈ ℂ
2	1	nex 1800	. . 3 ⊢ ¬ ∃𝑦(+∞ei‘𝑦) ∈ ℂ
3		elin 3967	. . . . . 6 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ∞))
4		df-bj-inftyexpi 37208	. . . . . . . . . . 11 ⊢ +∞ei = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
5	4	funmpt2 6605	. . . . . . . . . 10 ⊢ Fun +∞ei
6		elrnrexdm 7109	. . . . . . . . . 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 3076	. . . . . . . . 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 37213	. . . . . . . 8 ⊢ ℂ∞ = ran +∞ei
11	9, 10	eleq2s 2859	. . . . . . 7 ⊢ (𝑥 ∈ ℂ∞ → ∃𝑦 𝑥 = (+∞ei‘𝑦))
12	11	anim2i 617	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ∞) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei‘𝑦)))
13	3, 12	sylbi 217	. . . . 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 1861	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)) ↔ ∃𝑦(𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
16		19.41v 1949	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ) ↔ (∃𝑦 𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
17	15, 16	bitri 275	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)) ↔ (∃𝑦 𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
18	14, 17	sylbb2 238	. . . . 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 2829	. . . . . 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 1835	. . . 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 197	. 2 ⊢ ¬ 𝑥 ∈ (ℂ ∩ ℂ∞)
25	24	nel0 4354	1 ⊢ (ℂ ∩ ℂ∞) = ∅

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3070   ∩ cin 3950  ∅c0 4333  ⟨cop 4632  dom cdm 5685  ran crn 5686  Fun wfun 6555  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  -cneg 11493  (,]cioc 13388  πcpi 16102  +∞eⁱcinftyexpi 37207  ℂ_∞cccinfty 37212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-cnex 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-c 11161  df-bj-inftyexpi 37208  df-bj-ccinfty 37213

This theorem is referenced by: (None)