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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 PPoints
isside.2 PLines
isside.3 ss
isside.4 Ibg
isside.5
hpd.1.1
Assertion
Ref Expression
hpd

Proof of Theorem hpd
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 isside.1 . . . . . 6 PPoints
2 isside.2 . . . . . 6 PLines
3 isside.4 . . . . . . 7 Ibg
43isibg1a 26214 . . . . . 6 Ig
5 isside.5 . . . . . 6
61, 2, 4, 5gltpntl2 26176 . . . . 5
7 neq0 3478 . . . . 5
86, 7sylibr 203 . . . 4
98intnand 882 . . 3
10 eqcom 2298 . . . 4
11 sssu 25244 . . . 4
1210, 11bitri 240 . . 3
139, 12sylnibr 296 . 2
14 df-ne 2461 . 2
1513, 14sylibr 203 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358  wex 1531   wceq 1632   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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