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.

Step	Hyp	Ref	Expression
1		rpnnen 14061	. . . 4 |- RR ~~ ~P NN
2		cpnnen 14063	. . . 4 |- CC ~~ ~P NN
3	1, 2	entr4i 7530	. . 3 |- RR ~~ CC
4		bren 7483	. . 3 |- ( RR ~~ CC <-> E. a a : RR -1-1-onto-> CC )
5	3, 4	mpbi 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 ) )
9	7, 8	mpan2 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 )
12	10, 11	syl6bb 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 ) )
13	9, 12	syl 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 ) )
14	6, 13	mpbii 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
16	15, 15	xpex 6542	. . . . . 6 |- ( CC X. CC ) e. _V
17	16	inex2 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 ) )
19	17, 18	spcev 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 )
20	14, 19	syl 17	. . 3 |- ( a : RR -1-1-onto-> CC -> E. x x Or CC )
21	20	exlimiv 1743	. 2 |- ( E. a a : RR -1-1-onto-> CC -> E. x x Or CC )
22	5, 21	ax-mp 5	1 |- E. x x Or CC

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