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Theorem exmid01 4023
Description: Excluded middle is equivalent to saying any subset of  { (/) } is either  (/) or  { (/) }. (Contributed by BJ and Jim Kingdon, 18-Jun-2022.)
Assertion
Ref Expression
exmid01  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )

Proof of Theorem exmid01
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-exmid 4021 . 2  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
2 df-dc 781 . . . . 5  |-  (DECID  (/)  e.  x  <->  (
(/)  e.  x  \/  -.  (/)  e.  x ) )
3 orcom 682 . . . . . 6  |-  ( (
(/)  e.  x  \/  -.  (/)  e.  x )  <-> 
( -.  (/)  e.  x  \/  (/)  e.  x ) )
4 simpll 496 . . . . . . . . . . . . . 14  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  x  C_ 
{ (/) } )
5 simpr 108 . . . . . . . . . . . . . 14  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  y  e.  x )
64, 5sseldd 3024 . . . . . . . . . . . . 13  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  y  e.  { (/) } )
7 velsn 3458 . . . . . . . . . . . . 13  |-  ( y  e.  { (/) }  <->  y  =  (/) )
86, 7sylib 120 . . . . . . . . . . . 12  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  y  =  (/) )
98, 5eqeltrrd 2165 . . . . . . . . . . 11  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  (/)  e.  x
)
10 simplr 497 . . . . . . . . . . 11  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  -.  (/) 
e.  x )
119, 10pm2.65da 622 . . . . . . . . . 10  |-  ( ( x  C_  { (/) }  /\  -.  (/)  e.  x )  ->  -.  y  e.  x )
1211eq0rdv 3324 . . . . . . . . 9  |-  ( ( x  C_  { (/) }  /\  -.  (/)  e.  x )  ->  x  =  (/) )
1312ex 113 . . . . . . . 8  |-  ( x 
C_  { (/) }  ->  ( -.  (/)  e.  x  ->  x  =  (/) ) )
14 noel 3288 . . . . . . . . 9  |-  -.  (/)  e.  (/)
15 eleq2 2151 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( (/)  e.  x  <->  (/)  e.  (/) ) )
1614, 15mtbiri 635 . . . . . . . 8  |-  ( x  =  (/)  ->  -.  (/)  e.  x
)
1713, 16impbid1 140 . . . . . . 7  |-  ( x 
C_  { (/) }  ->  ( -.  (/)  e.  x  <->  x  =  (/) ) )
18 elex2 2635 . . . . . . . . . 10  |-  ( (/)  e.  x  ->  E. z 
z  e.  x )
19 sssnm 3593 . . . . . . . . . 10  |-  ( E. z  z  e.  x  ->  ( x  C_  { (/) }  <-> 
x  =  { (/) } ) )
2018, 19syl 14 . . . . . . . . 9  |-  ( (/)  e.  x  ->  ( x 
C_  { (/) }  <->  x  =  { (/) } ) )
2120biimpcd 157 . . . . . . . 8  |-  ( x 
C_  { (/) }  ->  (
(/)  e.  x  ->  x  =  { (/) } ) )
22 0ex 3958 . . . . . . . . . 10  |-  (/)  e.  _V
2322snid 3470 . . . . . . . . 9  |-  (/)  e.  { (/)
}
24 eleq2 2151 . . . . . . . . 9  |-  ( x  =  { (/) }  ->  (
(/)  e.  x  <->  (/)  e.  { (/)
} ) )
2523, 24mpbiri 166 . . . . . . . 8  |-  ( x  =  { (/) }  ->  (/)  e.  x )
2621, 25impbid1 140 . . . . . . 7  |-  ( x 
C_  { (/) }  ->  (
(/)  e.  x  <->  x  =  { (/) } ) )
2717, 26orbi12d 742 . . . . . 6  |-  ( x 
C_  { (/) }  ->  ( ( -.  (/)  e.  x  \/  (/)  e.  x )  <-> 
( x  =  (/)  \/  x  =  { (/) } ) ) )
283, 27syl5bb 190 . . . . 5  |-  ( x 
C_  { (/) }  ->  ( ( (/)  e.  x  \/  -.  (/)  e.  x )  <-> 
( x  =  (/)  \/  x  =  { (/) } ) ) )
292, 28syl5bb 190 . . . 4  |-  ( x 
C_  { (/) }  ->  (DECID  (/)  e.  x  <->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
3029pm5.74i 178 . . 3  |-  ( ( x  C_  { (/) }  -> DECID  (/)  e.  x
)  <->  ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
3130albii 1404 . 2  |-  ( A. x ( x  C_  {
(/) }  -> DECID  (/)  e.  x )  <->  A. x ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
321, 31bitri 182 1  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438    C_ wss 2997   (/)c0 3284   {csn 3441  EXMIDwem 4020
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-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3957
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-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447  df-exmid 4021
This theorem is referenced by:  exmidpw  6604  exmidomni  6777  ss1oel2o  11545  exmidsbthrlem  11569
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