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Theorem abvor0dc 3286
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 777 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 id 19 . . . . 5  |-  ( ph  ->  ph )
3 vex 2614 . . . . . 6  |-  x  e. 
_V
43a1i 9 . . . . 5  |-  ( ph  ->  x  e.  _V )
52, 42thd 173 . . . 4  |-  ( ph  ->  ( ph  <->  x  e.  _V ) )
65abbi1dv 2202 . . 3  |-  ( ph  ->  { x  |  ph }  =  _V )
7 id 19 . . . . 5  |-  ( -. 
ph  ->  -.  ph )
8 noel 3272 . . . . . 6  |-  -.  x  e.  (/)
98a1i 9 . . . . 5  |-  ( -. 
ph  ->  -.  x  e.  (/) )
107, 92falsed 651 . . . 4  |-  ( -. 
ph  ->  ( ph  <->  x  e.  (/) ) )
1110abbi1dv 2202 . . 3  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
126, 11orim12i 709 . 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 662  DECID wdc 776    = wceq 1285    e. wcel 1434   {cab 2069   _Vcvv 2611   (/)c0 3268
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 2065
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613  df-dif 2985  df-nul 3269
This theorem is referenced by: (None)
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