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Theorem exmidel 4034
Description: Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.)
Assertion
Ref Expression
exmidel  |-  (EXMID  <->  A. x A. yDECID  x  e.  y )
Distinct variable group:    x, y

Proof of Theorem exmidel
StepHypRef Expression
1 exmidexmid 4031 . . 3  |-  (EXMID  -> DECID  x  e.  y
)
21alrimivv 1803 . 2  |-  (EXMID  ->  A. x A. yDECID  x  e.  y )
3 0ex 3966 . . . 4  |-  (/)  e.  _V
4 eleq1 2150 . . . . . 6  |-  ( x  =  (/)  ->  ( x  e.  y  <->  (/)  e.  y ) )
54dcbid 786 . . . . 5  |-  ( x  =  (/)  ->  (DECID  x  e.  y  <-> DECID  (/) 
e.  y ) )
65albidv 1752 . . . 4  |-  ( x  =  (/)  ->  ( A. yDECID  x  e.  y  <->  A. yDECID  (/)  e.  y ) )
73, 6spcv 2712 . . 3  |-  ( A. x A. yDECID  x  e.  y  ->  A. yDECID  (/)  e.  y )
8 exmid0el 4033 . . 3  |-  (EXMID  <->  A. yDECID  (/)  e.  y )
97, 8sylibr 132 . 2  |-  ( A. x A. yDECID  x  e.  y  -> EXMID )
102, 9impbii 124 1  |-  (EXMID  <->  A. x A. yDECID  x  e.  y )
Colors of variables: wff set class
Syntax hints:    <-> wb 103  DECID wdc 780   A.wal 1287    = wceq 1289    e. wcel 1438   (/)c0 3286  EXMIDwem 4029
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-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-exmid 4030
This theorem is referenced by: (None)
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