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Theorem cnso 12525
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 rpnnen 12505 . . . 4  |-  RR  ~~  ~P NN
2 cpnnen 12507 . . . 4  |-  CC  ~~  ~P NN
31, 2entr4i 6918 . . 3  |-  RR  ~~  CC
4 bren 6871 . . 3  |-  ( RR 
~~  CC  <->  E. a  a : RR -1-1-onto-> CC )
53, 4mpbi 199 . 2  |-  E. a 
a : RR -1-1-onto-> CC
6 ltso 8903 . . . . 5  |-  <  Or  RR
7 eqid 2283 . . . . . . 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 5848 . . . . . . 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 ) )
97, 8mpan2 652 . . . . . 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 5845 . . . . . . 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 4754 . . . . . . 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 )
1210, 11syl6bb 252 . . . . . 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 ) )
139, 12syl 15 . . . . 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 ) )
146, 13mpbii 202 . . . 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 8818 . . . . . . 7  |-  CC  e.  _V
1615, 15xpex 4801 . . . . . 6  |-  ( CC 
X.  CC )  e. 
_V
1716inex2 4156 . . . . 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 4333 . . . . 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 ) )
1917, 18spcev 2875 . . . 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 )
2014, 19syl 15 . . 3  |-  ( a : RR -1-1-onto-> CC  ->  E. x  x  Or  CC )
2120exlimiv 1666 . 2  |-  ( E. a  a : RR -1-1-onto-> CC  ->  E. x  x  Or  CC )
225, 21ax-mp 8 1  |-  E. x  x  Or  CC
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623   E.wrex 2544    i^i cin 3151   ~Pcpw 3625   class class class wbr 4023   {copab 4076    Or wor 4313    X. cxp 4687   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256    ~~ cen 6860   CCcc 8735   RRcr 8736    < clt 8867   NNcn 9746
This theorem is referenced by:  aannenlem3  19710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159
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