Theorem bj-ccinftydisj 37540

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 37537	. . . 4 ⊢ ¬ (+∞ei‘𝑦) ∈ ℂ
2	1	nex 1802	. . 3 ⊢ ¬ ∃𝑦(+∞ei‘𝑦) ∈ ℂ
3		elin 3906	. . . . . 6 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ∞))
4		df-bj-inftyexpi 37534	. . . . . . . . . . 11 ⊢ +∞ei = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
5	4	funmpt2 6529	. . . . . . . . . 10 ⊢ Fun +∞ei
6		elrnrexdm 7033	. . . . . . . . . 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 3068	. . . . . . . . 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 37539	. . . . . . . 8 ⊢ ℂ∞ = ran +∞ei
11	9, 10	eleq2s 2855	. . . . . . 7 ⊢ (𝑥 ∈ ℂ∞ → ∃𝑦 𝑥 = (+∞ei‘𝑦))
12	11	anim2i 618	. . . . . 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 1863	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)) ↔ ∃𝑦(𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
16		19.41v 1951	. . . . . . 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 2825	. . . . . 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 1837	. . . 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 4295	1 ⊢ (ℂ ∩ ℂ∞) = ∅

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062   ∩ cin 3889  ∅c0 4274  ⟨cop 4574  dom cdm 5622  ran crn 5623  Fun wfun 6484  ‘cfv 6490  (class class class)co 7358  ℂcc 11025  -cneg 11367  (,]cioc 13288  πcpi 16020  +∞eⁱcinftyexpi 37533  ℂ_∞cccinfty 37538

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680  ax-reg 9498  ax-cnex 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-fv 6498  df-c 11033  df-bj-inftyexpi 37534  df-bj-ccinfty 37539

This theorem is referenced by: (None)