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Theorem ordtriexmidlem2 4272
Description: Lemma for decidability and ordinals. The set  { x  e.  { (/)
}  |  ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4273 or weak linearity in ordsoexmid 4313) 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 3262 . . 3  |-  -.  (/)  e.  (/)
2 eleq2 2143 . . 3  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  ( (/)  e.  { x  e.  { (/) }  |  ph } 
<->  (/)  e.  (/) ) )
31, 2mtbiri 633 . 2  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  -.  (/)  e.  { x  e.  { (/) }  |  ph } )
4 0ex 3913 . . . 4  |-  (/)  e.  _V
54snid 3433 . . 3  |-  (/)  e.  { (/)
}
6 biidd 170 . . . 4  |-  ( x  =  (/)  ->  ( ph  <->  ph ) )
76elrab3 2751 . . 3  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { x  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
85, 7ax-mp 7 . 2  |-  ( (/)  e.  { x  e.  { (/)
}  |  ph }  <->  ph )
93, 8sylnib 634 1  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   {crab 2353   (/)c0 3258   {csn 3406
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-nul 3912
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-v 2604  df-dif 2976  df-nul 3259  df-sn 3412
This theorem is referenced by:  ordtriexmid  4273  ordtri2orexmid  4274  ontr2exmid  4276  onsucsssucexmid  4278  ordsoexmid  4313  0elsucexmid  4316  ordpwsucexmid  4321
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