Theorem cnso 13832

Description: The complex numbers can be linearly ordered. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Assertion

Ref	Expression
cnso	|- E. x x Or CC

Proof of Theorem cnso

Dummy variables a b c d e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpnnen 13812	. . . 4 |- RR ~~ ~P NN
2		cpnnen 13814	. . . 4 |- CC ~~ ~P NN
3	1, 2	entr4i 7564	. . 3 |- RR ~~ CC
4		bren 7517	. . 3 |- ( RR ~~ CC <-> E. a a : RR -1-1-onto-> CC )
5	3, 4	mpbi 208	. 2 |- E. a a : RR -1-1-onto-> CC
6		ltso 9656	. . . . 5 |- < Or RR
7		eqid 2462	. . . . . . 7 |- { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } = { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) }
8		f1oiso 6228	. . . . . . 7 |- ( ( a : RR -1-1-onto-> CC /\ { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } = { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } ) -> a Isom < , { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } ( RR , CC ) )
9	7, 8	mpan2 671	. . . . . 6 |- ( a : RR -1-1-onto-> CC -> a Isom < , { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } ( RR , CC ) )
10		isoso 6225	. . . . . . 7 |- ( a Isom < , { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } ( RR , CC ) -> ( < Or RR <-> { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } Or CC ) )
11		soinxp 5058	. . . . . . 7 |- ( { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } Or CC <-> ( { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } i^i ( CC X. CC ) ) Or CC )
12	10, 11	syl6bb 261	. . . . . 6 |- ( a Isom < , { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } ( RR , CC ) -> ( < Or RR <-> ( { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } i^i ( CC X. CC ) ) Or CC ) )
13	9, 12	syl 16	. . . . 5 |- ( a : RR -1-1-onto-> CC -> ( < Or RR <-> ( { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } i^i ( CC X. CC ) ) Or CC ) )
14	6, 13	mpbii 211	. . . 4 |- ( a : RR -1-1-onto-> CC -> ( { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } i^i ( CC X. CC ) ) Or CC )
15		cnex 9564	. . . . . . 7 |- CC e. _V
16	15, 15	xpex 6706	. . . . . 6 |- ( CC X. CC ) e. _V
17	16	inex2 4584	. . . . 5 |- ( { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } i^i ( CC X. CC ) ) e. _V
18		soeq1 4814	. . . . 5 |- ( x = ( { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } i^i ( CC X. CC ) ) -> ( x Or CC <-> ( { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } i^i ( CC X. CC ) ) Or CC ) )
19	17, 18	spcev 3200	. . . 4 |- ( ( { <. b , c >. | E. d e. RR E. e e. RR ( ( b = ( a ` d ) /\ c = ( a ` e ) ) /\ d < e ) } i^i ( CC X. CC ) ) Or CC -> E. x x Or CC )
20	14, 19	syl 16	. . 3 |- ( a : RR -1-1-onto-> CC -> E. x x Or CC )
21	20	exlimiv 1693	. 2 |- ( E. a a : RR -1-1-onto-> CC -> E. x x Or CC )
22	5, 21	ax-mp 5	1 |- E. x x Or CC

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1374   E.wex 1591   E.wrex 2810   i^i cin 3470   ~Pcpw 4005   class class class wbr 4442   {copab 4499   Or wor 4794   X. cxp 4992   -1-1-onto->wf1o 5580   ` cfv 5581   Isom wiso 5582   ~~ cen 7505   CCcc 9481   RRcr 9482   < clt 9619   NNcn 10527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-oi 7926  df-card 8311  df-acn 8314  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-q 11174  df-rp 11212  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460

This theorem is referenced by:  aannenlem3  22455