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Theorem abvor0dc 3386
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 820 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
2 id 19 . . . . 5  |-  ( ph  ->  ph )
3 vex 2689 . . . . . 6  |-  x  e. 
_V
43a1i 9 . . . . 5  |-  ( ph  ->  x  e.  _V )
52, 42thd 174 . . . 4  |-  ( ph  ->  ( ph  <->  x  e.  _V ) )
65abbi1dv 2259 . . 3  |-  ( ph  ->  { x  |  ph }  =  _V )
7 id 19 . . . . 5  |-  ( -. 
ph  ->  -.  ph )
8 noel 3367 . . . . . 6  |-  -.  x  e.  (/)
98a1i 9 . . . . 5  |-  ( -. 
ph  ->  -.  x  e.  (/) )
107, 92falsed 691 . . . 4  |-  ( -. 
ph  ->  ( ph  <->  x  e.  (/) ) )
1110abbi1dv 2259 . . 3  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
126, 11orim12i 748 . 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 697  DECID wdc 819    = wceq 1331    e. wcel 1480   {cab 2125   _Vcvv 2686   (/)c0 3363
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-nul 3364
This theorem is referenced by: (None)
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