ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  exmid0el Unicode version

Theorem exmid0el 4178
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 4170 . . 3  |-  (EXMID  -> DECID  (/)  e.  x )
21alrimiv 1861 . 2  |-  (EXMID  ->  A. xDECID  (/)  e.  x
)
3 ax-1 6 . . . 4  |-  (DECID  (/)  e.  x  ->  ( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
43alimi 1442 . . 3  |-  ( A. xDECID  (/) 
e.  x  ->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
5 df-exmid 4169 . . 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 824   A.wal 1340    e. wcel 2135    C_ wss 3112   (/)c0 3405   {csn 3571  EXMIDwem 4168
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-nul 4103  ax-pow 4148
This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rab 2451  df-v 2724  df-dif 3114  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-exmid 4169
This theorem is referenced by:  exmidel  4179
  Copyright terms: Public domain W3C validator