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Theorem dcextest 4463
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 804 . . . 4  |-  (DECID  { x  |  ph }  e.  _V  ->  ( { x  | 
ph }  e.  _V  \/  -.  { x  | 
ph }  e.  _V ) )
31, 2ax-mp 5 . . 3  |-  ( { x  |  ph }  e.  _V  \/  -.  {
x  |  ph }  e.  _V )
4 vprc 4028 . . . . . . 7  |-  -.  _V  e.  _V
5 df-v 2660 . . . . . . . . 9  |-  _V  =  { x  |  x  =  x }
6 equid 1660 . . . . . . . . . . 11  |-  x  =  x
7 pm5.1im 172 . . . . . . . . . . 11  |-  ( x  =  x  ->  ( ph  ->  ( x  =  x  <->  ph ) ) )
86, 7ax-mp 5 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  x  <->  ph ) )
98abbidv 2233 . . . . . . . . 9  |-  ( ph  ->  { x  |  x  =  x }  =  { x  |  ph }
)
105, 9syl5req 2161 . . . . . . . 8  |-  ( ph  ->  { x  |  ph }  =  _V )
1110eleq1d 2184 . . . . . . 7  |-  ( ph  ->  ( { x  | 
ph }  e.  _V  <->  _V  e.  _V ) )
124, 11mtbiri 647 . . . . . 6  |-  ( ph  ->  -.  { x  | 
ph }  e.  _V )
1312con2i 599 . . . . 5  |-  ( { x  |  ph }  e.  _V  ->  -.  ph )
14 vex 2661 . . . . . . . . . 10  |-  y  e. 
_V
15 biidd 171 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ph 
<-> 
ph ) )
1614, 15elab 2800 . . . . . . . . 9  |-  ( y  e.  { x  | 
ph }  <->  ph )
1716notbii 640 . . . . . . . 8  |-  ( -.  y  e.  { x  |  ph }  <->  -.  ph )
1817biimpri 132 . . . . . . 7  |-  ( -. 
ph  ->  -.  y  e.  { x  |  ph }
)
1918eq0rdv 3375 . . . . . 6  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
20 0ex 4023 . . . . . 6  |-  (/)  e.  _V
2119, 20syl6eqel 2206 . . . . 5  |-  ( -. 
ph  ->  { x  | 
ph }  e.  _V )
2213, 21impbii 125 . . . 4  |-  ( { x  |  ph }  e.  _V  <->  -.  ph )
2322notbii 640 . . . 4  |-  ( -. 
{ x  |  ph }  e.  _V  <->  -.  -.  ph )
2422, 23orbi12i 736 . . 3  |-  ( ( { x  |  ph }  e.  _V  \/  -.  { x  |  ph }  e.  _V )  <->  ( -.  ph  \/  -.  -.  ph ) )
253, 24mpbi 144 . 2  |-  ( -. 
ph  \/  -.  -.  ph )
26 df-dc 803 . 2  |-  (DECID  -.  ph  <->  ( -.  ph  \/  -.  -.  ph ) )
2725, 26mpbir 145 1  |- DECID  -.  ph
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ wo 680  DECID wdc 802    e. wcel 1463   {cab 2101   _Vcvv 2658   (/)c0 3331
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022
This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332
This theorem is referenced by: (None)
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