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Theorem ncolne2 23715
Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.)
Hypotheses
Ref Expression
tglineelsb2.p  |-  B  =  ( Base `  G
)
tglineelsb2.i  |-  I  =  (Itv `  G )
tglineelsb2.l  |-  L  =  (LineG `  G )
tglineelsb2.g  |-  ( ph  ->  G  e. TarskiG )
ncolne.x  |-  ( ph  ->  X  e.  B )
ncolne.y  |-  ( ph  ->  Y  e.  B )
ncolne.z  |-  ( ph  ->  Z  e.  B )
ncolne.2  |-  ( ph  ->  -.  ( X  e.  ( Y L Z )  \/  Y  =  Z ) )
Assertion
Ref Expression
ncolne2  |-  ( ph  ->  X  =/=  Z )

Proof of Theorem ncolne2
StepHypRef Expression
1 tglineelsb2.p . 2  |-  B  =  ( Base `  G
)
2 tglineelsb2.i . 2  |-  I  =  (Itv `  G )
3 tglineelsb2.l . 2  |-  L  =  (LineG `  G )
4 tglineelsb2.g . 2  |-  ( ph  ->  G  e. TarskiG )
5 ncolne.x . 2  |-  ( ph  ->  X  e.  B )
6 ncolne.z . 2  |-  ( ph  ->  Z  e.  B )
7 ncolne.y . 2  |-  ( ph  ->  Y  e.  B )
8 ncolne.2 . . 3  |-  ( ph  ->  -.  ( X  e.  ( Y L Z )  \/  Y  =  Z ) )
94adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z L Y ) )  ->  G  e. TarskiG )
107adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z L Y ) )  ->  Y  e.  B )
116adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z L Y ) )  ->  Z  e.  B )
125adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z L Y ) )  ->  X  e.  B )
13 pm2.46 398 . . . . . . . . . 10  |-  ( -.  ( X  e.  ( Y L Z )  \/  Y  =  Z )  ->  -.  Y  =  Z )
148, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  -.  Y  =  Z )
1514neqned 2665 . . . . . . . 8  |-  ( ph  ->  Y  =/=  Z )
1615adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z L Y ) )  ->  Y  =/=  Z )
17 simpr 461 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z L Y ) )  ->  X  e.  ( Z L Y ) )
181, 2, 3, 9, 10, 11, 12, 16, 17lncom 23711 . . . . . 6  |-  ( (
ph  /\  X  e.  ( Z L Y ) )  ->  X  e.  ( Y L Z ) )
1918ex 434 . . . . 5  |-  ( ph  ->  ( X  e.  ( Z L Y )  ->  X  e.  ( Y L Z ) ) )
20 eqcom 2471 . . . . . . 7  |-  ( Z  =  Y  <->  Y  =  Z )
2120biimpi 194 . . . . . 6  |-  ( Z  =  Y  ->  Y  =  Z )
2221a1i 11 . . . . 5  |-  ( ph  ->  ( Z  =  Y  ->  Y  =  Z ) )
2319, 22orim12d 835 . . . 4  |-  ( ph  ->  ( ( X  e.  ( Z L Y )  \/  Z  =  Y )  ->  ( X  e.  ( Y L Z )  \/  Y  =  Z ) ) )
2423con3d 133 . . 3  |-  ( ph  ->  ( -.  ( X  e.  ( Y L Z )  \/  Y  =  Z )  ->  -.  ( X  e.  ( Z L Y )  \/  Z  =  Y ) ) )
258, 24mpd 15 . 2  |-  ( ph  ->  -.  ( X  e.  ( Z L Y )  \/  Z  =  Y ) )
261, 2, 3, 4, 5, 6, 7, 25ncolne1 23714 1  |-  ( ph  ->  X  =/=  Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657   ` cfv 5581  (class class class)co 6277   Basecbs 14481  TarskiGcstrkg 23548  Itvcitv 23555  LineGclng 23556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-trkgc 23567  df-trkgb 23568  df-trkgcb 23569  df-trkg 23573
This theorem is referenced by:  midexlem  23772
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