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Theorem abvor0dc 3306
Description: The class builder of a decidable proposition not containing the abstraction variable is either the universal class or the empty set. (Contributed by Jim Kingdon, 1-Aug-2018.)
Assertion
Ref Expression
abvor0dc  |-  (DECID  ph  ->  ( { x  |  ph }  =  _V  \/  { x  |  ph }  =  (/) ) )
Distinct variable group:    ph, x

Proof of Theorem abvor0dc
StepHypRef Expression
1 df-dc 781 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 id 19 . . . . 5  |-  ( ph  ->  ph )
3 vex 2622 . . . . . 6  |-  x  e. 
_V
43a1i 9 . . . . 5  |-  ( ph  ->  x  e.  _V )
52, 42thd 173 . . . 4  |-  ( ph  ->  ( ph  <->  x  e.  _V ) )
65abbi1dv 2207 . . 3  |-  ( ph  ->  { x  |  ph }  =  _V )
7 id 19 . . . . 5  |-  ( -. 
ph  ->  -.  ph )
8 noel 3290 . . . . . 6  |-  -.  x  e.  (/)
98a1i 9 . . . . 5  |-  ( -. 
ph  ->  -.  x  e.  (/) )
107, 92falsed 653 . . . 4  |-  ( -. 
ph  ->  ( ph  <->  x  e.  (/) ) )
1110abbi1dv 2207 . . 3  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
126, 11orim12i 711 . 2  |-  ( (
ph  \/  -.  ph )  ->  ( { x  | 
ph }  =  _V  \/  { x  |  ph }  =  (/) ) )
131, 12sylbi 119 1  |-  (DECID  ph  ->  ( { x  |  ph }  =  _V  \/  { x  |  ph }  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 664  DECID wdc 780    = wceq 1289    e. wcel 1438   {cab 2074   _Vcvv 2619   (/)c0 3286
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
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-nul 3287
This theorem is referenced by: (None)
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