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Theorem ordtriexmidlem2 4537
Description: Lemma for decidability and ordinals. The set  { x  e.  { (/)
}  |  ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4538 or weak linearity in ordsoexmid 4579) 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 3441 . . 3  |-  -.  (/)  e.  (/)
2 eleq2 2253 . . 3  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  ( (/)  e.  { x  e.  { (/) }  |  ph } 
<->  (/)  e.  (/) ) )
31, 2mtbiri 676 . 2  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  -.  (/)  e.  { x  e.  { (/) }  |  ph } )
4 0ex 4145 . . . 4  |-  (/)  e.  _V
54snid 3638 . . 3  |-  (/)  e.  { (/)
}
6 biidd 172 . . . 4  |-  ( x  =  (/)  ->  ( ph  <->  ph ) )
76elrab3 2909 . . 3  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { x  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
85, 7ax-mp 5 . 2  |-  ( (/)  e.  { x  e.  { (/)
}  |  ph }  <->  ph )
93, 8sylnib 677 1  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105    = wceq 1364    e. wcel 2160   {crab 2472   (/)c0 3437   {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-nul 4144
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-dif 3146  df-nul 3438  df-sn 3613
This theorem is referenced by:  ordtriexmid  4538  ontriexmidim  4539  ordtri2orexmid  4540  ontr2exmid  4542  onsucsssucexmid  4544  ordsoexmid  4579  0elsucexmid  4582  ordpwsucexmid  4587
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