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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.
StepHypRef Expression
1 rpnnen 13611 . . . 4  |-  RR  ~~  ~P NN
2 cpnnen 13613 . . . 4  |-  CC  ~~  ~P NN
31, 2entr4i 7466 . . 3  |-  RR  ~~  CC
4 bren 7419 . . 3  |-  ( RR 
~~  CC  <->  E. a  a : RR -1-1-onto-> CC )
53, 4mpbi 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 ) )
97, 8mpan2 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 )
1210, 11syl6bb 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 ) )
139, 12syl 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 ) )
146, 13mpbii 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
1615, 15xpex 6608 . . . . . 6  |-  ( CC 
X.  CC )  e. 
_V
1716inex2 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 ) )
1917, 18spcev 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 )
2014, 19syl 16 . . 3  |-  ( a : RR -1-1-onto-> CC  ->  E. x  x  Or  CC )
2120exlimiv 1689 . 2  |-  ( E. a  a : RR -1-1-onto-> CC  ->  E. x  x  Or  CC )
225, 21ax-mp 5 1  |-  E. x  x  Or  CC
Colors of variables: wff setvar class
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
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