Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rpnnen3 Unicode version

Theorem rpnnen3 26448
Description: Dedekind cut injection of  RR into  ~P QQ. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Assertion
Ref Expression
rpnnen3  |-  RR  ~<_  ~P QQ

Proof of Theorem rpnnen3
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qex 10417 . . 3  |-  QQ  e.  _V
21pwex 4272 . 2  |-  ~P QQ  e.  _V
3 ssrab2 3334 . . . . 5  |-  { c  e.  QQ  |  c  <  a }  C_  QQ
41elpw2 4254 . . . . 5  |-  ( { c  e.  QQ  | 
c  <  a }  e.  ~P QQ  <->  { c  e.  QQ  |  c  < 
a }  C_  QQ )
53, 4mpbir 200 . . . 4  |-  { c  e.  QQ  |  c  <  a }  e.  ~P QQ
65a1i 10 . . 3  |-  ( a  e.  RR  ->  { c  e.  QQ  |  c  <  a }  e.  ~P QQ )
7 lttri2 8991 . . . . . 6  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( a  =/=  b  <->  ( a  <  b  \/  b  <  a ) ) )
8 rpnnen3lem 26447 . . . . . . . 8  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  a  <  b
)  ->  { c  e.  QQ  |  c  < 
a }  =/=  {
c  e.  QQ  | 
c  <  b }
)
9 rpnnen3lem 26447 . . . . . . . . . 10  |-  ( ( ( b  e.  RR  /\  a  e.  RR )  /\  b  <  a
)  ->  { c  e.  QQ  |  c  < 
b }  =/=  {
c  e.  QQ  | 
c  <  a }
)
109ancom1s 780 . . . . . . . . 9  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  b  <  a
)  ->  { c  e.  QQ  |  c  < 
b }  =/=  {
c  e.  QQ  | 
c  <  a }
)
1110necomd 2604 . . . . . . . 8  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  b  <  a
)  ->  { c  e.  QQ  |  c  < 
a }  =/=  {
c  e.  QQ  | 
c  <  b }
)
128, 11jaodan 760 . . . . . . 7  |-  ( ( ( a  e.  RR  /\  b  e.  RR )  /\  ( a  < 
b  \/  b  < 
a ) )  ->  { c  e.  QQ  |  c  <  a }  =/=  { c  e.  QQ  |  c  < 
b } )
1312ex 423 . . . . . 6  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( ( a  < 
b  \/  b  < 
a )  ->  { c  e.  QQ  |  c  <  a }  =/=  { c  e.  QQ  | 
c  <  b }
) )
147, 13sylbid 206 . . . . 5  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( a  =/=  b  ->  { c  e.  QQ  |  c  <  a }  =/=  { c  e.  QQ  |  c  < 
b } ) )
1514necon4d 2584 . . . 4  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( { c  e.  QQ  |  c  < 
a }  =  {
c  e.  QQ  | 
c  <  b }  ->  a  =  b ) )
16 breq2 4106 . . . . 5  |-  ( a  =  b  ->  (
c  <  a  <->  c  <  b ) )
1716rabbidv 2856 . . . 4  |-  ( a  =  b  ->  { c  e.  QQ  |  c  <  a }  =  { c  e.  QQ  |  c  <  b } )
1815, 17impbid1 194 . . 3  |-  ( ( a  e.  RR  /\  b  e.  RR )  ->  ( { c  e.  QQ  |  c  < 
a }  =  {
c  e.  QQ  | 
c  <  b }  <->  a  =  b ) )
196, 18dom2 6989 . 2  |-  ( ~P QQ  e.  _V  ->  RR  ~<_  ~P QQ )
202, 19ax-mp 8 1  |-  RR  ~<_  ~P QQ
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   {crab 2623   _Vcvv 2864    C_ wss 3228   ~Pcpw 3701   class class class wbr 4102    ~<_ cdom 6946   RRcr 8823    < clt 8954   QQcq 10405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-n0 10055  df-z 10114  df-uz 10320  df-q 10406
  Copyright terms: Public domain W3C validator