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Theorem pwtrufal 13952
Description: A subset of the singleton  { (/) } cannot be anything other than  (/) or  { (/) }. Removing the double negation would change the meaning, as seen at exmid01 4182. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4180), 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 527 . . . . 5  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  -.  A  =  { (/) } )
2 simpll 524 . . . . . . . 8  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  A  C_  { (/) } )
3 simpl 108 . . . . . . . . . . . 12  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  A  C_  { (/) } )
43sselda 3147 . . . . . . . . . . 11  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  x  e.  {
(/) } )
5 elsni 3599 . . . . . . . . . . 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 2248 . . . . . . . . 9  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  (/)  e.  A
)
98snssd 3723 . . . . . . . 8  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  { (/) }  C_  A )
102, 9eqssd 3164 . . . . . . 7  |-  ( ( ( A  C_  { (/) }  /\  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )  /\  x  e.  A
)  ->  A  =  { (/) } )
1110ex 114 . . . . . 6  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  ( x  e.  A  ->  A  =  { (/) } ) )
1211exlimdv 1812 . . . . 5  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  ( E. x  x  e.  A  ->  A  =  { (/) } ) )
131, 12mtod 658 . . . 4  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  -.  E. x  x  e.  A )
14 notm0 3434 . . . 4  |-  ( -. 
E. x  x  e.  A  <->  A  =  (/) )
1513, 14sylib 121 . . 3  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  A  =  (/) )
16 simprl 526 . . 3  |-  ( ( A  C_  { (/) }  /\  ( -.  A  =  (/) 
/\  -.  A  =  { (/) } ) )  ->  -.  A  =  (/) )
1715, 16pm2.65da 656 . 2  |-  ( A 
C_  { (/) }  ->  -.  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )
18 ioran 747 . 2  |-  ( -.  ( A  =  (/)  \/  A  =  { (/) } )  <->  ( -.  A  =  (/)  /\  -.  A  =  { (/) } ) )
1917, 18sylnibr 672 1  |-  ( A 
C_  { (/) }  ->  -. 
-.  ( A  =  (/)  \/  A  =  { (/)
} ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 703    = wceq 1348   E.wex 1485    e. wcel 2141    C_ wss 3121   (/)c0 3414   {csn 3581
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3587
This theorem is referenced by:  pwle2  13953
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