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Theorem exmid01 4241
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 variable  y is distinct from all other variables.
StepHypRef Expression
1 df-exmid 4238 . 2  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
2 df-dc 836 . . . . 5  |-  (DECID  (/)  e.  x  <->  (
(/)  e.  x  \/  -.  (/)  e.  x ) )
3 orcom 729 . . . . . 6  |-  ( (
(/)  e.  x  \/  -.  (/)  e.  x )  <-> 
( -.  (/)  e.  x  \/  (/)  e.  x ) )
4 simpll 527 . . . . . . . . . . . . . 14  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  x  C_ 
{ (/) } )
5 simpr 110 . . . . . . . . . . . . . 14  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  y  e.  x )
64, 5sseldd 3193 . . . . . . . . . . . . 13  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  y  e.  { (/) } )
7 velsn 3649 . . . . . . . . . . . . 13  |-  ( y  e.  { (/) }  <->  y  =  (/) )
86, 7sylib 122 . . . . . . . . . . . 12  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  y  =  (/) )
98, 5eqeltrrd 2282 . . . . . . . . . . 11  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  (/)  e.  x
)
10 simplr 528 . . . . . . . . . . 11  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  -.  (/) 
e.  x )
119, 10pm2.65da 662 . . . . . . . . . 10  |-  ( ( x  C_  { (/) }  /\  -.  (/)  e.  x )  ->  -.  y  e.  x )
1211eq0rdv 3504 . . . . . . . . 9  |-  ( ( x  C_  { (/) }  /\  -.  (/)  e.  x )  ->  x  =  (/) )
1312ex 115 . . . . . . . 8  |-  ( x 
C_  { (/) }  ->  ( -.  (/)  e.  x  ->  x  =  (/) ) )
14 noel 3463 . . . . . . . . 9  |-  -.  (/)  e.  (/)
15 eleq2 2268 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( (/)  e.  x  <->  (/)  e.  (/) ) )
1614, 15mtbiri 676 . . . . . . . 8  |-  ( x  =  (/)  ->  -.  (/)  e.  x
)
1713, 16impbid1 142 . . . . . . 7  |-  ( x 
C_  { (/) }  ->  ( -.  (/)  e.  x  <->  x  =  (/) ) )
18 ss1o0el1 4240 . . . . . . 7  |-  ( x 
C_  { (/) }  ->  (
(/)  e.  x  <->  x  =  { (/) } ) )
1917, 18orbi12d 794 . . . . . 6  |-  ( x 
C_  { (/) }  ->  ( ( -.  (/)  e.  x  \/  (/)  e.  x )  <-> 
( x  =  (/)  \/  x  =  { (/) } ) ) )
203, 19bitrid 192 . . . . 5  |-  ( x 
C_  { (/) }  ->  ( ( (/)  e.  x  \/  -.  (/)  e.  x )  <-> 
( x  =  (/)  \/  x  =  { (/) } ) ) )
212, 20bitrid 192 . . . 4  |-  ( x 
C_  { (/) }  ->  (DECID  (/)  e.  x  <->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
2221pm5.74i 180 . . 3  |-  ( ( x  C_  { (/) }  -> DECID  (/)  e.  x
)  <->  ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
2322albii 1492 . 2  |-  ( A. x ( x  C_  {
(/) }  -> DECID  (/)  e.  x )  <->  A. x ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
241, 23bitri 184 1  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709  DECID wdc 835   A.wal 1370    = wceq 1372    e. wcel 2175    C_ wss 3165   (/)c0 3459   {csn 3632  EXMIDwem 4237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-nul 4169
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178  df-nul 3460  df-sn 3638  df-exmid 4238
This theorem is referenced by:  exmid1dc  4243  exmidn0m  4244  exmidsssn  4245  exmidpw  7004  exmidpweq  7005  exmidomni  7243  ss1oel2o  15890  exmidsbthrlem  15923  sbthom  15927
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