Theorem cnso 14376

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		ltso 9732	. . . 4 |- < Or RR
2		eqid 2471	. . . . . 6 |- { <. 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 ) }
3		f1oiso 6260	. . . . . 6 |- ( ( 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 ) )
4	2, 3	mpan2 685	. . . . 5 |- ( 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 ) )
5		isoso 6257	. . . . . 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 ) } Or CC ) )
6		soinxp 4904	. . . . . 6 |- ( { <. 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 )
7	5, 6	syl6bb 269	. . . . 5 |- ( 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 ) )
8	4, 7	syl 17	. . . 4 |- ( 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 ) )
9	1, 8	mpbii 216	. . 3 |- ( 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 )
10		cnex 9638	. . . . . 6 |- CC e. _V
11	10, 10	xpex 6614	. . . . 5 |- ( CC X. CC ) e. _V
12	11	inex2 4538	. . . 4 |- ( { <. 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
13		soeq1 4779	. . . 4 |- ( 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 ) )
14	12, 13	spcev 3127	. . 3 |- ( ( { <. 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 )
15	9, 14	syl 17	. 2 |- ( a : RR -1-1-onto-> CC -> E. x x Or CC )
16		rpnnen 14356	. . . 4 |- RR ~~ ~P NN
17		cpnnen 14358	. . . 4 |- CC ~~ ~P NN
18	16, 17	entr4i 7644	. . 3 |- RR ~~ CC
19		bren 7596	. . 3 |- ( RR ~~ CC <-> E. a a : RR -1-1-onto-> CC )
20	18, 19	mpbi 213	. 2 |- E. a a : RR -1-1-onto-> CC
21	15, 20	exlimiiv 1785	1 |- E. x x Or CC

Syntax hints:   <-> wb 189   /\ wa 376   = wceq 1452   E.wex 1671   E.wrex 2757   i^i cin 3389   ~Pcpw 3942   class class class wbr 4395   {copab 4453   Or wor 4759   X. cxp 4837   -1-1-onto->wf1o 5588   ` cfv 5589   Isom wiso 5590   ~~ cen 7584   CCcc 9555   RRcr 9556   < clt 9693   NNcn 10631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830

This theorem is referenced by:  aannenlem3  23365