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Theorem hpd 26272
Description: Halfplanes are distinct. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Hypotheses
Ref Expression
isside.1  |-  P  =  (PPoints `  G )
isside.2  |-  L  =  (PLines `  G )
isside.3  |-  .~  =  ( (ss `  G ) `
 M )
isside.4  |-  ( ph  ->  G  e. Ibg )
isside.5  |-  ( ph  ->  M  e.  L )
hpd.1.1  |-  ( ph  ->  A  e.  ( P 
\  M ) )
Assertion
Ref Expression
hpd  |-  ( ph  ->  [ A ]  .~  =/=  ( ( P  \  M )  \  [ A ]  .~  )
)

Proof of Theorem hpd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isside.1 . . . . . 6  |-  P  =  (PPoints `  G )
2 isside.2 . . . . . 6  |-  L  =  (PLines `  G )
3 isside.4 . . . . . . 7  |-  ( ph  ->  G  e. Ibg )
43isibg1a 26214 . . . . . 6  |-  ( ph  ->  G  e. Ig )
5 isside.5 . . . . . 6  |-  ( ph  ->  M  e.  L )
61, 2, 4, 5gltpntl2 26176 . . . . 5  |-  ( ph  ->  E. x  x  e.  ( P  \  M
) )
7 neq0 3478 . . . . 5  |-  ( -.  ( P  \  M
)  =  (/)  <->  E. x  x  e.  ( P  \  M ) )
86, 7sylibr 203 . . . 4  |-  ( ph  ->  -.  ( P  \  M )  =  (/) )
98intnand 882 . . 3  |-  ( ph  ->  -.  ( [ A ]  .~  =  (/)  /\  ( P  \  M )  =  (/) ) )
10 eqcom 2298 . . . 4  |-  ( [ A ]  .~  =  ( ( P  \  M )  \  [ A ]  .~  )  <->  ( ( P  \  M
)  \  [ A ]  .~  )  =  [ A ]  .~  )
11 sssu 25244 . . . 4  |-  ( ( ( P  \  M
)  \  [ A ]  .~  )  =  [ A ]  .~  <->  ( [ A ]  .~  =  (/) 
/\  ( P  \  M )  =  (/) ) )
1210, 11bitri 240 . . 3  |-  ( [ A ]  .~  =  ( ( P  \  M )  \  [ A ]  .~  )  <->  ( [ A ]  .~  =  (/)  /\  ( P 
\  M )  =  (/) ) )
139, 12sylnibr 296 . 2  |-  ( ph  ->  -.  [ A ]  .~  =  ( ( P  \  M )  \  [ A ]  .~  )
)
14 df-ne 2461 . 2  |-  ( [ A ]  .~  =/=  ( ( P  \  M )  \  [ A ]  .~  )  <->  -. 
[ A ]  .~  =  ( ( P 
\  M )  \  [ A ]  .~  )
)
1513, 14sylibr 203 1  |-  ( ph  ->  [ A ]  .~  =/=  ( ( P  \  M )  \  [ A ]  .~  )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   (/)c0 3468   ` cfv 5271   [cec 6674  PPointscpoints 26159  PLinescplines 26161  Ibgcibg 26210  sscsas 26265
This theorem is referenced by:  bhp3  26280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-ig2 26164  df-ibg2 26212
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