Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hpd Unicode version

Theorem hpd 26169
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 26111 . . . . . 6  |-  ( ph  ->  G  e. Ig )
5 isside.5 . . . . . 6  |-  ( ph  ->  M  e.  L )
61, 2, 4, 5gltpntl2 26073 . . . . 5  |-  ( ph  ->  E. x  x  e.  ( P  \  M
) )
7 neq0 3465 . . . . 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 2285 . . . 4  |-  ( [ A ]  .~  =  ( ( P  \  M )  \  [ A ]  .~  )  <->  ( ( P  \  M
)  \  [ A ]  .~  )  =  [ A ]  .~  )
11 sssu 25141 . . . 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 2448 . 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 1528    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   (/)c0 3455   ` cfv 5255   [cec 6658  PPointscpoints 26056  PLinescplines 26058  Ibgcibg 26107  sscsas 26162
This theorem is referenced by:  bhp3  26177
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-ig2 26061  df-ibg2 26109
  Copyright terms: Public domain W3C validator