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Theorem pwntru 4090
Description: A slight strengthening of pwtrufal 12994. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
Assertion
Ref Expression
pwntru  |-  ( ( A  C_  { (/) }  /\  A  =/=  { (/) } )  ->  A  =  (/) )

Proof of Theorem pwntru
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . 4  |-  ( ( A  C_  { (/) }  /\  A  =/=  { (/) } )  ->  A  =/=  { (/)
} )
21neneqd 2304 . . 3  |-  ( ( A  C_  { (/) }  /\  A  =/=  { (/) } )  ->  -.  A  =  { (/) } )
3 simpll 501 . . . . . 6  |-  ( ( ( A  C_  { (/) }  /\  A  =/=  { (/)
} )  /\  x  e.  A )  ->  A  C_ 
{ (/) } )
4 simpl 108 . . . . . . . . . 10  |-  ( ( A  C_  { (/) }  /\  A  =/=  { (/) } )  ->  A  C_  { (/) } )
54sselda 3065 . . . . . . . . 9  |-  ( ( ( A  C_  { (/) }  /\  A  =/=  { (/)
} )  /\  x  e.  A )  ->  x  e.  { (/) } )
6 elsni 3513 . . . . . . . . 9  |-  ( x  e.  { (/) }  ->  x  =  (/) )
75, 6syl 14 . . . . . . . 8  |-  ( ( ( A  C_  { (/) }  /\  A  =/=  { (/)
} )  /\  x  e.  A )  ->  x  =  (/) )
8 simpr 109 . . . . . . . 8  |-  ( ( ( A  C_  { (/) }  /\  A  =/=  { (/)
} )  /\  x  e.  A )  ->  x  e.  A )
97, 8eqeltrrd 2193 . . . . . . 7  |-  ( ( ( A  C_  { (/) }  /\  A  =/=  { (/)
} )  /\  x  e.  A )  ->  (/)  e.  A
)
109snssd 3633 . . . . . 6  |-  ( ( ( A  C_  { (/) }  /\  A  =/=  { (/)
} )  /\  x  e.  A )  ->  { (/) } 
C_  A )
113, 10eqssd 3082 . . . . 5  |-  ( ( ( A  C_  { (/) }  /\  A  =/=  { (/)
} )  /\  x  e.  A )  ->  A  =  { (/) } )
1211ex 114 . . . 4  |-  ( ( A  C_  { (/) }  /\  A  =/=  { (/) } )  ->  ( x  e.  A  ->  A  =  { (/) } ) )
1312exlimdv 1773 . . 3  |-  ( ( A  C_  { (/) }  /\  A  =/=  { (/) } )  ->  ( E. x  x  e.  A  ->  A  =  { (/) } ) )
142, 13mtod 635 . 2  |-  ( ( A  C_  { (/) }  /\  A  =/=  { (/) } )  ->  -.  E. x  x  e.  A )
15 notm0 3351 . 2  |-  ( -. 
E. x  x  e.  A  <->  A  =  (/) )
1614, 15sylib 121 1  |-  ( ( A  C_  { (/) }  /\  A  =/=  { (/) } )  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1314   E.wex 1451    e. wcel 1463    =/= wne 2283    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-ne 2284  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501
This theorem is referenced by:  exmid1dc  4091  exmid1stab  12997
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