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Theorem cnso 14081
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 14061 . . . 4  |-  RR  ~~  ~P NN
2 cpnnen 14063 . . . 4  |-  CC  ~~  ~P NN
31, 2entr4i 7530 . . 3  |-  RR  ~~  CC
4 bren 7483 . . 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 9616 . . . . 5  |-  <  Or  RR
7 eqid 2402 . . . . . . 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 6186 . . . . . . 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 669 . . . . . 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 6183 . . . . . . 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 5007 . . . . . . 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 17 . . . . 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 9523 . . . . . . 7  |-  CC  e.  _V
1615, 15xpex 6542 . . . . . 6  |-  ( CC 
X.  CC )  e. 
_V
1716inex2 4535 . . . . 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 4762 . . . . 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 3150 . . . 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 17 . . 3  |-  ( a : RR -1-1-onto-> CC  ->  E. x  x  Or  CC )
2120exlimiv 1743 . 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 367    = wceq 1405   E.wex 1633   E.wrex 2754    i^i cin 3412   ~Pcpw 3954   class class class wbr 4394   {copab 4451    Or wor 4742    X. cxp 4940   -1-1-onto->wf1o 5524   ` cfv 5525    Isom wiso 5526    ~~ cen 7471   CCcc 9440   RRcr 9441    < clt 9578   NNcn 10496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-2o 7088  df-oadd 7091  df-omul 7092  df-er 7268  df-map 7379  df-pm 7380  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-oi 7889  df-card 8272  df-acn 8275  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-q 11146  df-rp 11184  df-ico 11506  df-icc 11507  df-fz 11644  df-fzo 11768  df-fl 11879  df-seq 12062  df-exp 12121  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-limsup 13350  df-clim 13367  df-rlim 13368  df-sum 13565
This theorem is referenced by:  aannenlem3  22910
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