Theorem cnso 13631

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 13611	. . . 4 |- RR ~~ ~P NN
2		cpnnen 13613	. . . 4 |- CC ~~ ~P NN
3	1, 2	entr4i 7466	. . 3 |- RR ~~ CC
4		bren 7419	. . 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 9556	. . . . 5 |- < Or RR
7		eqid 2451	. . . . . . 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 6141	. . . . . . 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 6138	. . . . . . 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 5001	. . . . . . 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 9464	. . . . . . 7 |- CC e. _V
16	15, 15	xpex 6608	. . . . . 6 |- ( CC X. CC ) e. _V
17	16	inex2 4532	. . . . 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 4758	. . . . 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 3160	. . . 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 1689	. 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 1370   E.wex 1587   E.wrex 2796   i^i cin 3425   ~Pcpw 3958   class class class wbr 4390   {copab 4447   Or wor 4738   X. cxp 4936   -1-1-onto->wf1o 5515   ` cfv 5516   Isom wiso 5517   ~~ cen 7407   CCcc 9381   RRcr 9382   < clt 9519   NNcn 10423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-omul 7025  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-acn 8213  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-q 11055  df-rp 11093  df-ico 11407  df-icc 11408  df-fz 11539  df-fzo 11650  df-fl 11743  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-limsup 13051  df-clim 13068  df-rlim 13069  df-sum 13266

This theorem is referenced by:  aannenlem3  21912