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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.
StepHypRef 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 ) )
42, 3mpan2 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 )
75, 6syl6bb 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 ) )
84, 7syl 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 ) )
91, 8mpbii 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
1110, 10xpex 6614 . . . . 5  |-  ( CC 
X.  CC )  e. 
_V
1211inex2 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 ) )
1412, 13spcev 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 )
159, 14syl 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
1816, 17entr4i 7644 . . 3  |-  RR  ~~  CC
19 bren 7596 . . 3  |-  ( RR 
~~  CC  <->  E. a  a : RR -1-1-onto-> CC )
2018, 19mpbi 213 . 2  |-  E. a 
a : RR -1-1-onto-> CC
2115, 20exlimiiv 1785 1  |-  E. x  x  Or  CC
Colors of variables: wff setvar class
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
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