Theorem bj-ccinftydisj 33100

Description: The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-ccinftydisj	|- ( CC i^i CCinfty ) = (/)

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 33097	. . . 4 |- -. (inftyexpi ` y ) e. CC
2	1	nex 1731	. . 3 |- -. E. y (inftyexpi ` y ) e. CC
3		elin 3796	. . . . . 6 |- ( x e. ( CC i^i CCinfty ) <-> ( x e. CC /\ x e. CCinfty ) )
4		df-bj-inftyexpi 33094	. . . . . . . . . . 11 |- inftyexpi = ( z e. ( -u pi (,] pi ) |-> <. z , CC >. )
5	4	funmpt2 5927	. . . . . . . . . 10 |- Fun inftyexpi
6		elrnrexdm 6363	. . . . . . . . . 10 |- ( Fun inftyexpi -> ( x e. ran inftyexpi -> E. y e. dom inftyexpi x = (inftyexpi ` y ) ) )
7	5, 6	ax-mp 5	. . . . . . . . 9 |- ( x e. ran inftyexpi -> E. y e. dom inftyexpi x = (inftyexpi ` y ) )
8		rexex 3002	. . . . . . . . 9 |- ( E. y e. dom inftyexpi x = (inftyexpi ` y ) -> E. y x = (inftyexpi ` y ) )
9	7, 8	syl 17	. . . . . . . 8 |- ( x e. ran inftyexpi -> E. y x = (inftyexpi ` y ) )
10		df-bj-ccinfty 33099	. . . . . . . 8 |- CCinfty = ran inftyexpi
11	9, 10	eleq2s 2719	. . . . . . 7 |- ( x e. CCinfty -> E. y x = (inftyexpi ` y ) )
12	11	anim2i 593	. . . . . 6 |- ( ( x e. CC /\ x e. CCinfty ) -> ( x e. CC /\ E. y x = (inftyexpi ` y ) ) )
13	3, 12	sylbi 207	. . . . 5 |- ( x e. ( CC i^i CCinfty ) -> ( x e. CC /\ E. y x = (inftyexpi ` y ) ) )
14		ancom 466	. . . . . 6 |- ( ( x e. CC /\ E. y x = (inftyexpi ` y ) ) <-> ( E. y x = (inftyexpi ` y ) /\ x e. CC ) )
15		exancom 1787	. . . . . . 7 |- ( E. y ( x e. CC /\ x = (inftyexpi ` y ) ) <-> E. y ( x = (inftyexpi ` y ) /\ x e. CC ) )
16		19.41v 1914	. . . . . . 7 |- ( E. y ( x = (inftyexpi ` y ) /\ x e. CC ) <-> ( E. y x = (inftyexpi ` y ) /\ x e. CC ) )
17	15, 16	bitri 264	. . . . . 6 |- ( E. y ( x e. CC /\ x = (inftyexpi ` y ) ) <-> ( E. y x = (inftyexpi ` y ) /\ x e. CC ) )
18	14, 17	sylbb2 228	. . . . 5 |- ( ( x e. CC /\ E. y x = (inftyexpi ` y ) ) -> E. y ( x e. CC /\ x = (inftyexpi ` y ) ) )
19	13, 18	syl 17	. . . 4 |- ( x e. ( CC i^i CCinfty ) -> E. y ( x e. CC /\ x = (inftyexpi ` y ) ) )
20		eleq1 2689	. . . . . 6 |- ( x = (inftyexpi ` y ) -> ( x e. CC <-> (inftyexpi ` y ) e. CC ) )
21	20	biimpac 503	. . . . 5 |- ( ( x e. CC /\ x = (inftyexpi ` y ) ) -> (inftyexpi ` y ) e. CC )
22	21	eximi 1762	. . . 4 |- ( E. y ( x e. CC /\ x = (inftyexpi ` y ) ) -> E. y (inftyexpi ` y ) e. CC )
23	19, 22	syl 17	. . 3 |- ( x e. ( CC i^i CCinfty ) -> E. y (inftyexpi ` y ) e. CC )
24	2, 23	mto 188	. 2 |- -. x e. ( CC i^i CCinfty )
25	24	nel0 3932	1 |- ( CC i^i CCinfty ) = (/)

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   i^i cin 3573   (/)c0 3915   <.cop 4183   dom cdm 5114   ran crn 5115   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   CCcc 9934   -ucneg 10267   (,]cioc 12176   picpi 14797  inftyexpi cinftyexpi 33093  CCinftycccinfty 33098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-c 9942  df-bj-inftyexpi 33094  df-bj-ccinfty 33099

This theorem is referenced by: (None)