ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordpwsucexmid Structured version   Unicode version

Theorem ordpwsucexmid 4230
Description: The subset in ordpwsucss 4227 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.)
Hypothesis
Ref Expression
ordpwsucexmid.1  On  suc  ~P  i^i  On
Assertion
Ref Expression
ordpwsucexmid
Distinct variable group:   ,

Proof of Theorem ordpwsucexmid
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 0elpw 3891 . . . . 5  (/)  ~P {  { (/) }  |  }
2 0elon 4078 . . . . 5  (/)  On
3 elin 3103 . . . . 5  (/)  ~P {  { (/) }  |  }  i^i  On  (/)  ~P {  { (/) }  |  }  (/)  On
41, 2, 3mpbir2an 837 . . . 4  (/)  ~P { 
{ (/) }  |  }  i^i  On
5 ordtriexmidlem 4192 . . . . 5  {  { (/) }  |  }  On
6 suceq 4088 . . . . . . 7  { 
{ (/) }  |  }  suc 
suc  {  { (/)
}  |  }
7 pweq 3337 . . . . . . . 8  { 
{ (/) }  |  }  ~P  ~P { 
{ (/) }  |  }
87ineq1d 3114 . . . . . . 7  { 
{ (/) }  |  }  ~P  i^i  On  ~P { 
{ (/) }  |  }  i^i  On
96, 8eqeq12d 2036 . . . . . 6  { 
{ (/) }  |  }  suc  ~P  i^i  On  suc  {  { (/) }  |  }  ~P {  { (/)
}  |  }  i^i  On
10 ordpwsucexmid.1 . . . . . 6  On  suc  ~P  i^i  On
119, 10vtoclri 2605 . . . . 5  {  { (/) }  |  }  On  suc 
{  { (/)
}  |  }  ~P {  { (/) }  |  }  i^i  On
125, 11ax-mp 7 . . . 4  suc  {  { (/) }  |  }  ~P {  { (/)
}  |  }  i^i  On
134, 12eleqtrri 2095 . . 3  (/)  suc  {  { (/) }  |  }
14 elsuci 4089 . . 3  (/)  suc  { 
{ (/) }  |  }  (/)  {  { (/) }  |  }  (/)  {  { (/) }  |  }
1513, 14ax-mp 7 . 2  (/)  {  { (/)
}  |  }  (/)  {  { (/) }  |  }
16 0ex 3858 . . . . . 6  (/)  _V
1716snid 3377 . . . . 5  (/)  { (/)
}
18 biidd 161 . . . . . 6  (/)
1918elrab3 2676 . . . . 5  (/)  { (/) }  (/)  {  { (/)
}  |  }
2017, 19ax-mp 7 . . . 4  (/)  {  { (/)
}  |  }
2120biimpi 113 . . 3  (/)  {  { (/)
}  |  }
22 ordtriexmidlem2 4193 . . . 4  {  { (/) }  |  }  (/)
2322eqcoms 2025 . . 3  (/)  {  { (/)
}  |  }
2421, 23orim12i 663 . 2  (/)  { 
{ (/) }  |  }  (/)  {  { (/) }  |  }
2515, 24ax-mp 7 1
Colors of variables: wff set class
Syntax hints:   wn 3   wb 98   wo 616   wceq 1228   wcel 1374  wral 2284   {crab 2288    i^i cin 2893   (/)c0 3201   ~Pcpw 3334   {csn 3350   Oncon0 4049   suc csuc 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator