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Theorem exmidel 4189
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 4180 . . 3  |-  (EXMID  -> DECID  x  e.  y
)
21alrimivv 1868 . 2  |-  (EXMID  ->  A. x A. yDECID  x  e.  y )
3 0ex 4114 . . . 4  |-  (/)  e.  _V
4 eleq1 2233 . . . . . 6  |-  ( x  =  (/)  ->  ( x  e.  y  <->  (/)  e.  y ) )
54dcbid 833 . . . . 5  |-  ( x  =  (/)  ->  (DECID  x  e.  y  <-> DECID  (/) 
e.  y ) )
65albidv 1817 . . . 4  |-  ( x  =  (/)  ->  ( A. yDECID  x  e.  y  <->  A. yDECID  (/)  e.  y ) )
73, 6spcv 2824 . . 3  |-  ( A. x A. yDECID  x  e.  y  ->  A. yDECID  (/)  e.  y )
8 exmid0el 4188 . . 3  |-  (EXMID  <->  A. yDECID  (/)  e.  y )
97, 8sylibr 133 . 2  |-  ( A. x A. yDECID  x  e.  y  -> EXMID )
102, 9impbii 125 1  |-  (EXMID  <->  A. x A. yDECID  x  e.  y )
Colors of variables: wff set class
Syntax hints:    <-> wb 104  DECID wdc 829   A.wal 1346    = wceq 1348    e. wcel 2141   (/)c0 3414  EXMIDwem 4178
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-exmid 4179
This theorem is referenced by: (None)
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