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Theorem exmid0el 4097
Description: Excluded middle is equivalent to decidability of  (/) being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.)
Assertion
Ref Expression
exmid0el  |-  (EXMID  <->  A. xDECID  (/)  e.  x
)

Proof of Theorem exmid0el
StepHypRef Expression
1 exmidexmid 4090 . . 3  |-  (EXMID  -> DECID  (/)  e.  x )
21alrimiv 1830 . 2  |-  (EXMID  ->  A. xDECID  (/)  e.  x
)
3 ax-1 6 . . . 4  |-  (DECID  (/)  e.  x  ->  ( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
43alimi 1416 . . 3  |-  ( A. xDECID  (/) 
e.  x  ->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
5 df-exmid 4089 . . 3  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
64, 5sylibr 133 . 2  |-  ( A. xDECID  (/) 
e.  x  -> EXMID )
72, 6impbii 125 1  |-  (EXMID  <->  A. xDECID  (/)  e.  x
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104  DECID wdc 804   A.wal 1314    e. wcel 1465    C_ wss 3041   (/)c0 3333   {csn 3497  EXMIDwem 4088
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-dc 805  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-exmid 4089
This theorem is referenced by:  exmidel  4098
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