Theorem bj-ccinftydisj 37717

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 37714	. . . 4 ⊢ ¬ (+∞ei‘𝑦) ∈ ℂ
2	1	nex 1823	. . 3 ⊢ ¬ ∃𝑦(+∞ei‘𝑦) ∈ ℂ
3		elin 3923	. . . . . 6 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ∞))
4		df-bj-inftyexpi 37711	. . . . . . . . . . 11 ⊢ +∞ei = (𝑧 ∈ (-π(,]π) ↦ ⟨𝑧, ℂ⟩)
5	4	funmpt2 6564	. . . . . . . . . 10 ⊢ Fun +∞ei
6		elrnrexdm 7074	. . . . . . . . . 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 3095	. . . . . . . . 9 ⊢ (∃𝑦 ∈ dom +∞ei𝑥 = (+∞ei‘𝑦) → ∃𝑦 𝑥 = (+∞ei‘𝑦))
9	7, 8	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ ran +∞ei → ∃𝑦 𝑥 = (+∞ei‘𝑦))
10		df-bj-ccinfty 37716	. . . . . . . 8 ⊢ ℂ∞ = ran +∞ei
11	9, 10	eleq2s 2883	. . . . . . 7 ⊢ (𝑥 ∈ ℂ∞ → ∃𝑦 𝑥 = (+∞ei‘𝑦))
12	11	anim2i 628	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ∞) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei‘𝑦)))
13	3, 12	sylbi 220	. . . . 5 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) → (𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei‘𝑦)))
14		ancom 465	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei‘𝑦)) ↔ (∃𝑦 𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
15		exancom 1884	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)) ↔ ∃𝑦(𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
16		19.41v 1972	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ) ↔ (∃𝑦 𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
17	15, 16	bitri 278	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)) ↔ (∃𝑦 𝑥 = (+∞ei‘𝑦) ∧ 𝑥 ∈ ℂ))
18	14, 17	sylbb2 241	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ ∃𝑦 𝑥 = (+∞ei‘𝑦)) → ∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)))
19	13, 18	syl 18	. . . 4 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) → ∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)))
20		eleq1 2853	. . . . . 6 ⊢ (𝑥 = (+∞ei‘𝑦) → (𝑥 ∈ ℂ ↔ (+∞ei‘𝑦) ∈ ℂ))
21	20	biimpac 483	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)) → (+∞ei‘𝑦) ∈ ℂ)
22	21	eximi 1858	. . . 4 ⊢ (∃𝑦(𝑥 ∈ ℂ ∧ 𝑥 = (+∞ei‘𝑦)) → ∃𝑦(+∞ei‘𝑦) ∈ ℂ)
23	19, 22	syl 18	. . 3 ⊢ (𝑥 ∈ (ℂ ∩ ℂ∞) → ∃𝑦(+∞ei‘𝑦) ∈ ℂ)
24	2, 23	mto 200	. 2 ⊢ ¬ 𝑥 ∈ (ℂ ∩ ℂ∞)
25	24	nel0 4310	1 ⊢ (ℂ ∩ ℂ∞) = ∅

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∃wrex 3089   ∩ cin 3906  ∅c0 4288  ⟨cop 4591  dom cdm 5652  ran crn 5653  Fun wfun 6519  ‘cfv 6525  (class class class)co 7400  ℂcc 11086  -cneg 11430  (,]cioc 13364  πcpi 16110  +∞eⁱcinftyexpi 37710  ℂ_∞cccinfty 37715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722  ax-reg 9542  ax-cnex 11144

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fn 6528  df-fv 6533  df-c 11094  df-bj-inftyexpi 37711  df-bj-ccinfty 37716

This theorem is referenced by: (None)