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Theorem ordtriexmidlem2 4502
Description: Lemma for decidability and ordinals. The set  { x  e.  { (/)
}  |  ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4503 or weak linearity in ordsoexmid 4544) 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 3418 . . 3  |-  -.  (/)  e.  (/)
2 eleq2 2234 . . 3  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  ( (/)  e.  { x  e.  { (/) }  |  ph } 
<->  (/)  e.  (/) ) )
31, 2mtbiri 670 . 2  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  -.  (/)  e.  { x  e.  { (/) }  |  ph } )
4 0ex 4114 . . . 4  |-  (/)  e.  _V
54snid 3612 . . 3  |-  (/)  e.  { (/)
}
6 biidd 171 . . . 4  |-  ( x  =  (/)  ->  ( ph  <->  ph ) )
76elrab3 2887 . . 3  |-  ( (/)  e.  { (/) }  ->  ( (/) 
e.  { x  e. 
{ (/) }  |  ph } 
<-> 
ph ) )
85, 7ax-mp 5 . 2  |-  ( (/)  e.  { x  e.  { (/)
}  |  ph }  <->  ph )
93, 8sylnib 671 1  |-  ( { x  e.  { (/) }  |  ph }  =  (/) 
->  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    = wceq 1348    e. wcel 2141   {crab 2452   (/)c0 3414   {csn 3581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4113
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-nul 3415  df-sn 3587
This theorem is referenced by:  ordtriexmid  4503  ontriexmidim  4504  ordtri2orexmid  4505  ontr2exmid  4507  onsucsssucexmid  4509  ordsoexmid  4544  0elsucexmid  4547  ordpwsucexmid  4552
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