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Theorem abvor0dc 3356
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 805 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 id 19 . . . . 5  |-  ( ph  ->  ph )
3 vex 2663 . . . . . 6  |-  x  e. 
_V
43a1i 9 . . . . 5  |-  ( ph  ->  x  e.  _V )
52, 42thd 174 . . . 4  |-  ( ph  ->  ( ph  <->  x  e.  _V ) )
65abbi1dv 2237 . . 3  |-  ( ph  ->  { x  |  ph }  =  _V )
7 id 19 . . . . 5  |-  ( -. 
ph  ->  -.  ph )
8 noel 3337 . . . . . 6  |-  -.  x  e.  (/)
98a1i 9 . . . . 5  |-  ( -. 
ph  ->  -.  x  e.  (/) )
107, 92falsed 676 . . . 4  |-  ( -. 
ph  ->  ( ph  <->  x  e.  (/) ) )
1110abbi1dv 2237 . . 3  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
126, 11orim12i 733 . 2  |-  ( (
ph  \/  -.  ph )  ->  ( { x  | 
ph }  =  _V  \/  { x  |  ph }  =  (/) ) )
131, 12sylbi 120 1  |-  (DECID  ph  ->  ( { x  |  ph }  =  _V  \/  { x  |  ph }  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 682  DECID wdc 804    = wceq 1316    e. wcel 1465   {cab 2103   _Vcvv 2660   (/)c0 3333
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-dc 805  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-nul 3334
This theorem is referenced by: (None)
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