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Theorem sgplpte21c 26238
Description: The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
sgplpte.1  |-  P  =  (PPoints `  G )
sgplpte.3  |-  S  =  ( seg `  G
)
sgplpte.4  |-  ( ph  ->  G  e. Ibg )
sgplpte.5  |-  ( ph  ->  X  e.  P )
sgplpte21c.2  |-  B  =  (btw `  G )
sgplpte21c.6  |-  ( ph  ->  Y  e.  P )
sgplpte21c.7  |-  ( ph  ->  X  =/=  Y )
Assertion
Ref Expression
sgplpte21c  |-  ( ph  ->  ( Z  e.  ( X S Y )  ->  ( Z  e.  P  /\  ( Z  e.  ( X B Y )  \/  Z  =  X  \/  Z  =  Y ) ) ) )

Proof of Theorem sgplpte21c
StepHypRef Expression
1 sgplpte.1 . . . . 5  |-  P  =  (PPoints `  G )
2 sgplpte.3 . . . . 5  |-  S  =  ( seg `  G
)
3 sgplpte.4 . . . . . 6  |-  ( ph  ->  G  e. Ibg )
43adantl 452 . . . . 5  |-  ( ( Z  e.  ( X S Y )  /\  ph )  ->  G  e. Ibg )
5 sgplpte.5 . . . . . 6  |-  ( ph  ->  X  e.  P )
65adantl 452 . . . . 5  |-  ( ( Z  e.  ( X S Y )  /\  ph )  ->  X  e.  P )
7 sgplpte21c.2 . . . . 5  |-  B  =  (btw `  G )
8 sgplpte21c.6 . . . . . 6  |-  ( ph  ->  Y  e.  P )
98adantl 452 . . . . 5  |-  ( ( Z  e.  ( X S Y )  /\  ph )  ->  Y  e.  P )
10 sgplpte21c.7 . . . . . 6  |-  ( ph  ->  X  =/=  Y )
1110adantl 452 . . . . 5  |-  ( ( Z  e.  ( X S Y )  /\  ph )  ->  X  =/=  Y )
12 elex 2809 . . . . . 6  |-  ( Z  e.  ( X S Y )  ->  Z  e.  _V )
1312adantr 451 . . . . 5  |-  ( ( Z  e.  ( X S Y )  /\  ph )  ->  Z  e.  _V )
141, 2, 4, 6, 7, 9, 11, 13sgplpte21b 26237 . . . 4  |-  ( ( Z  e.  ( X S Y )  /\  ph )  ->  ( Z  e.  ( X S Y )  <->  ( Z  e.  P  /\  ( Z  e.  ( X B Y )  \/  Z  =  X  \/  Z  =  Y ) ) ) )
1514biimpd 198 . . 3  |-  ( ( Z  e.  ( X S Y )  /\  ph )  ->  ( Z  e.  ( X S Y )  ->  ( Z  e.  P  /\  ( Z  e.  ( X B Y )  \/  Z  =  X  \/  Z  =  Y ) ) ) )
1615ex 423 . 2  |-  ( Z  e.  ( X S Y )  ->  ( ph  ->  ( Z  e.  ( X S Y )  ->  ( Z  e.  P  /\  ( Z  e.  ( X B Y )  \/  Z  =  X  \/  Z  =  Y ) ) ) ) )
1716pm2.43b 46 1  |-  ( ph  ->  ( Z  e.  ( X S Y )  ->  ( Z  e.  P  /\  ( Z  e.  ( X B Y )  \/  Z  =  X  \/  Z  =  Y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   ` cfv 5271  (class class class)co 5874  PPointscpoints 26159  btwcbtw 26209  Ibgcibg 26210   segcseg 26233
This theorem is referenced by:  sgplpte21e  26240
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-seg2 26234
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