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Theorem ordpwsucexmid 4246
Description: The subset in ordpwsucss 4243 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 3908 . . . . 5  (/)  ~P {  { (/) }  |  }
2 0elon 4095 . . . . 5  (/)  On
3 elin 3120 . . . . 5  (/)  ~P {  { (/) }  |  }  i^i  On  (/)  ~P {  { (/) }  |  }  (/)  On
41, 2, 3mpbir2an 848 . . . 4  (/)  ~P { 
{ (/) }  |  }  i^i  On
5 ordtriexmidlem 4208 . . . . 5  {  { (/) }  |  }  On
6 suceq 4105 . . . . . . 7  { 
{ (/) }  |  }  suc 
suc  {  { (/)
}  |  }
7 pweq 3354 . . . . . . . 8  { 
{ (/) }  |  }  ~P  ~P { 
{ (/) }  |  }
87ineq1d 3131 . . . . . . 7  { 
{ (/) }  |  }  ~P  i^i  On  ~P { 
{ (/) }  |  }  i^i  On
96, 8eqeq12d 2051 . . . . . 6  { 
{ (/) }  |  }  suc  ~P  i^i  On  suc  {  { (/) }  |  }  ~P {  { (/)
}  |  }  i^i  On
10 ordpwsucexmid.1 . . . . . 6  On  suc  ~P  i^i  On
119, 10vtoclri 2622 . . . . 5  {  { (/) }  |  }  On  suc 
{  { (/)
}  |  }  ~P {  { (/) }  |  }  i^i  On
125, 11ax-mp 7 . . . 4  suc  {  { (/) }  |  }  ~P {  { (/)
}  |  }  i^i  On
134, 12eleqtrri 2110 . . 3  (/)  suc  {  { (/) }  |  }
14 elsuci 4106 . . 3  (/)  suc  { 
{ (/) }  |  }  (/)  {  { (/) }  |  }  (/)  {  { (/) }  |  }
1513, 14ax-mp 7 . 2  (/)  {  { (/)
}  |  }  (/)  {  { (/) }  |  }
16 0ex 3875 . . . . . 6  (/)  _V
1716snid 3394 . . . . 5  (/)  { (/)
}
18 biidd 161 . . . . . 6  (/)
1918elrab3 2693 . . . . 5  (/)  { (/) }  (/)  {  { (/)
}  |  }
2017, 19ax-mp 7 . . . 4  (/)  {  { (/)
}  |  }
2120biimpi 113 . . 3  (/)  {  { (/)
}  |  }
22 ordtriexmidlem2 4209 . . . 4  {  { (/) }  |  }  (/)
2322eqcoms 2040 . . 3  (/)  {  { (/)
}  |  }
2421, 23orim12i 675 . 2  (/)  { 
{ (/) }  |  }  (/)  {  { (/) }  |  }
2515, 24ax-mp 7 1
Colors of variables: wff set class
Syntax hints:   wn 3   wb 98   wo 628   wceq 1242   wcel 1390  wral 2300   {crab 2304    i^i cin 2910   (/)c0 3218   ~Pcpw 3351   {csn 3367   Oncon0 4066   suc csuc 4068
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074
This theorem is referenced by: (None)
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