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Theorem pwtrufal 13003
Description: A subset of the singleton  { (/) } cannot be anything other than  (/) or  { (/) }. Removing the double negation would change the meaning, as seen at exmid01 4089. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4088), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.)
Assertion
Ref Expression
pwtrufal  |-  ( A 
C_  { (/) }  ->  -. 
-.  ( A  =  (/)  \/  A  =  { (/)
} ) )

Proof of Theorem pwtrufal
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simprr 504 . . . . 5  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  -.  A  =  { (/) } )
2 simpll 501 . . . . . . . 8  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  A  C_  { (/) } )
3 simpl 108 . . . . . . . . . . . 12  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  A  C_  { (/) } )
43sselda 3065 . . . . . . . . . . 11  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  x  e.  {
(/) } )
5 elsni 3513 . . . . . . . . . . 11  |-  ( x  e.  { (/) }  ->  x  =  (/) )
64, 5syl 14 . . . . . . . . . 10  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  x  =  (/) )
7 simpr 109 . . . . . . . . . 10  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  x  e.  A )
86, 7eqeltrrd 2193 . . . . . . . . 9  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  (/)  e.  A
)
98snssd 3633 . . . . . . . 8  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  { (/) }  C_  A )
102, 9eqssd 3082 . . . . . . 7  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  A  =  { (/) } )
1110ex 114 . . . . . 6  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  ( x  e.  A  ->  A  =  { (/) } ) )
1211exlimdv 1773 . . . . 5  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  ( E. x  x  e.  A  ->  A  =  { (/) } ) )
131, 12mtod 635 . . . 4  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  -.  E. x  x  e.  A )
14 notm0 3351 . . . 4  |-  ( -. 
E. x  x  e.  A  <->  A  =  (/) )
1513, 14sylib 121 . . 3  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  A  =  (/) )
16 simprl 503 . . 3  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  -.  A  =  (/) )
1715, 16pm2.65da 633 . 2  |-  ( A 
C_  { (/) }  ->  -.  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )
18 ioran 724 . 2  |-  ( -.  ( A  =  (/)  \/  A  =  { (/) } )  <->  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )
1917, 18sylnibr 649 1  |-  ( A 
C_  { (/) }  ->  -. 
-.  ( A  =  (/)  \/  A  =  { (/)
} ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 680    = wceq 1314   E.wex 1451    e. wcel 1463    C_ wss 3039   (/)c0 3331   {csn 3495
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
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  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
This theorem is referenced by:  pwle2  13004
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