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Theorem dcextest 4386
Description: If it is decidable whether  { x  | 
ph } is a set, then  -.  ph is decidable (where  x does not occur in 
ph). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition  -.  ph is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)
Hypothesis
Ref Expression
dcextest.ex  |- DECID  { x  |  ph }  e.  _V
Assertion
Ref Expression
dcextest  |- DECID  -.  ph
Distinct variable group:    ph, x

Proof of Theorem dcextest
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dcextest.ex . . . 4  |- DECID  { x  |  ph }  e.  _V
2 exmiddc 782 . . . 4  |-  (DECID  { x  |  ph }  e.  _V  ->  ( { x  | 
ph }  e.  _V  \/  -.  { x  | 
ph }  e.  _V ) )
31, 2ax-mp 7 . . 3  |-  ( { x  |  ph }  e.  _V  \/  -.  {
x  |  ph }  e.  _V )
4 vprc 3963 . . . . . . 7  |-  -.  _V  e.  _V
5 df-v 2621 . . . . . . . . 9  |-  _V  =  { x  |  x  =  x }
6 equid 1634 . . . . . . . . . . 11  |-  x  =  x
7 pm5.1im 171 . . . . . . . . . . 11  |-  ( x  =  x  ->  ( ph  ->  ( x  =  x  <->  ph ) ) )
86, 7ax-mp 7 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  x  <->  ph ) )
98abbidv 2205 . . . . . . . . 9  |-  ( ph  ->  { x  |  x  =  x }  =  { x  |  ph }
)
105, 9syl5req 2133 . . . . . . . 8  |-  ( ph  ->  { x  |  ph }  =  _V )
1110eleq1d 2156 . . . . . . 7  |-  ( ph  ->  ( { x  | 
ph }  e.  _V  <->  _V  e.  _V ) )
124, 11mtbiri 635 . . . . . 6  |-  ( ph  ->  -.  { x  | 
ph }  e.  _V )
1312con2i 592 . . . . 5  |-  ( { x  |  ph }  e.  _V  ->  -.  ph )
14 vex 2622 . . . . . . . . . 10  |-  y  e. 
_V
15 biidd 170 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ph 
<-> 
ph ) )
1614, 15elab 2758 . . . . . . . . 9  |-  ( y  e.  { x  | 
ph }  <->  ph )
1716notbii 629 . . . . . . . 8  |-  ( -.  y  e.  { x  |  ph }  <->  -.  ph )
1817biimpri 131 . . . . . . 7  |-  ( -. 
ph  ->  -.  y  e.  { x  |  ph }
)
1918eq0rdv 3324 . . . . . 6  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
20 0ex 3958 . . . . . 6  |-  (/)  e.  _V
2119, 20syl6eqel 2178 . . . . 5  |-  ( -. 
ph  ->  { x  | 
ph }  e.  _V )
2213, 21impbii 124 . . . 4  |-  ( { x  |  ph }  e.  _V  <->  -.  ph )
2322notbii 629 . . . 4  |-  ( -. 
{ x  |  ph }  e.  _V  <->  -.  -.  ph )
2422, 23orbi12i 716 . . 3  |-  ( ( { x  |  ph }  e.  _V  \/  -.  { x  |  ph }  e.  _V )  <->  ( -.  ph  \/  -.  -.  ph ) )
253, 24mpbi 143 . 2  |-  ( -. 
ph  \/  -.  -.  ph )
26 dftest 860 . 2  |-  (DECID  -.  ph  <->  ( -.  ph  \/  -.  -.  ph ) )
2725, 26mpbir 144 1  |- DECID  -.  ph
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ wo 664  DECID wdc 780    e. wcel 1438   {cab 2074   _Vcvv 2619   (/)c0 3284
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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285
This theorem is referenced by: (None)
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