ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordtriexmidlem2 Unicode version

Theorem ordtriexmidlem2 4337
Description: Lemma for decidability and ordinals. The set  { x  e.  { (/)
}  |  ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4338 or weak linearity in ordsoexmid 4378) with a proposition  ph. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.)
Assertion
Ref Expression
ordtriexmidlem2  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
Distinct variable group:    ph, x

Proof of Theorem ordtriexmidlem2
StepHypRef Expression
1 noel 3290 . . 3  |-  -.  (/)  e.  (/)
2 eleq2 2151 . . 3  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  ( (/)  e.  { x  e.  { (/) }  |  ph } 
<->  (/)  e.  (/) ) )
31, 2mtbiri 635 . 2  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  -.  (/)  e.  { x  e.  { (/) }  |  ph } )
4 0ex 3966 . . . 4  |-  (/)  e.  _V
54snid 3475 . . 3  |-  (/)  e.  { (/)
}
6 biidd 170 . . . 4  |-  ( x  =  (/)  ->  ( ph  <->  ph ) )
76elrab3 2772 . . 3  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { x  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
85, 7ax-mp 7 . 2  |-  ( (/)  e.  { x  e.  { (/)
}  |  ph }  <->  ph )
93, 8sylnib 636 1  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438   {crab 2363   (/)c0 3286   {csn 3446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3965
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-dif 3001  df-nul 3287  df-sn 3452
This theorem is referenced by:  ordtriexmid  4338  ordtri2orexmid  4339  ontr2exmid  4341  onsucsssucexmid  4343  ordsoexmid  4378  0elsucexmid  4381  ordpwsucexmid  4386
  Copyright terms: Public domain W3C validator