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Theorem exmid01 4089
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 4087 . 2  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
2 df-dc 803 . . . . 5  |-  (DECID  (/)  e.  x  <->  (
(/)  e.  x  \/  -.  (/)  e.  x ) )
3 orcom 700 . . . . . 6  |-  ( (
(/)  e.  x  \/  -.  (/)  e.  x )  <-> 
( -.  (/)  e.  x  \/  (/)  e.  x ) )
4 simpll 501 . . . . . . . . . . . . . 14  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  x  C_ 
{ (/) } )
5 simpr 109 . . . . . . . . . . . . . 14  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  y  e.  x )
64, 5sseldd 3066 . . . . . . . . . . . . 13  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  y  e.  { (/) } )
7 velsn 3512 . . . . . . . . . . . . 13  |-  ( y  e.  { (/) }  <->  y  =  (/) )
86, 7sylib 121 . . . . . . . . . . . 12  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  y  =  (/) )
98, 5eqeltrrd 2193 . . . . . . . . . . 11  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  (/)  e.  x
)
10 simplr 502 . . . . . . . . . . 11  |-  ( ( ( x  C_  { (/) }  /\  -.  (/)  e.  x
)  /\  y  e.  x )  ->  -.  (/) 
e.  x )
119, 10pm2.65da 633 . . . . . . . . . 10  |-  ( ( x  C_  { (/) }  /\  -.  (/)  e.  x )  ->  -.  y  e.  x )
1211eq0rdv 3375 . . . . . . . . 9  |-  ( ( x  C_  { (/) }  /\  -.  (/)  e.  x )  ->  x  =  (/) )
1312ex 114 . . . . . . . 8  |-  ( x 
C_  { (/) }  ->  ( -.  (/)  e.  x  ->  x  =  (/) ) )
14 noel 3335 . . . . . . . . 9  |-  -.  (/)  e.  (/)
15 eleq2 2179 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( (/)  e.  x  <->  (/)  e.  (/) ) )
1614, 15mtbiri 647 . . . . . . . 8  |-  ( x  =  (/)  ->  -.  (/)  e.  x
)
1713, 16impbid1 141 . . . . . . 7  |-  ( x 
C_  { (/) }  ->  ( -.  (/)  e.  x  <->  x  =  (/) ) )
18 elex2 2674 . . . . . . . . . 10  |-  ( (/)  e.  x  ->  E. z 
z  e.  x )
19 sssnm 3649 . . . . . . . . . 10  |-  ( E. z  z  e.  x  ->  ( x  C_  { (/) }  <-> 
x  =  { (/) } ) )
2018, 19syl 14 . . . . . . . . 9  |-  ( (/)  e.  x  ->  ( x 
C_  { (/) }  <->  x  =  { (/) } ) )
2120biimpcd 158 . . . . . . . 8  |-  ( x 
C_  { (/) }  ->  (
(/)  e.  x  ->  x  =  { (/) } ) )
22 0ex 4023 . . . . . . . . . 10  |-  (/)  e.  _V
2322snid 3524 . . . . . . . . 9  |-  (/)  e.  { (/)
}
24 eleq2 2179 . . . . . . . . 9  |-  ( x  =  { (/) }  ->  (
(/)  e.  x  <->  (/)  e.  { (/)
} ) )
2523, 24mpbiri 167 . . . . . . . 8  |-  ( x  =  { (/) }  ->  (/)  e.  x )
2621, 25impbid1 141 . . . . . . 7  |-  ( x 
C_  { (/) }  ->  (
(/)  e.  x  <->  x  =  { (/) } ) )
2717, 26orbi12d 765 . . . . . 6  |-  ( x 
C_  { (/) }  ->  ( ( -.  (/)  e.  x  \/  (/)  e.  x )  <-> 
( x  =  (/)  \/  x  =  { (/) } ) ) )
283, 27syl5bb 191 . . . . 5  |-  ( x 
C_  { (/) }  ->  ( ( (/)  e.  x  \/  -.  (/)  e.  x )  <-> 
( x  =  (/)  \/  x  =  { (/) } ) ) )
292, 28syl5bb 191 . . . 4  |-  ( x 
C_  { (/) }  ->  (DECID  (/)  e.  x  <->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
3029pm5.74i 179 . . 3  |-  ( ( x  C_  { (/) }  -> DECID  (/)  e.  x
)  <->  ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
3130albii 1429 . 2  |-  ( A. x ( x  C_  {
(/) }  -> DECID  (/)  e.  x )  <->  A. x ( x  C_  {
(/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
321, 31bitri 183 1  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680  DECID wdc 802   A.wal 1312    = wceq 1314   E.wex 1451    e. wcel 1463    C_ wss 3039   (/)c0 3331   {csn 3495  EXMIDwem 4086
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022
This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-exmid 4087
This theorem is referenced by:  exmid1dc  4091  exmidn0m  4092  exmidsssn  4093  exmidpw  6768  exmidomni  6980  ss1oel2o  12991  exmidsbthrlem  13019  sbthom  13023
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