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Theorem btwncolinear1 26005
Description: Betweenness implies colinearity. (Contributed by Scott Fenton, 7-Oct-2013.)
Assertion
Ref Expression
btwncolinear1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  A  Colinear  <. B ,  C >. ) )

Proof of Theorem btwncolinear1
StepHypRef Expression
1 3mix3 1129 . 2  |-  ( C 
Btwn  <. A ,  B >.  ->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) )
2 brcolinear 25995 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
31, 2syl5ibr 214 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  A  Colinear  <. B ,  C >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    \/ w3o 936    /\ w3a 937    e. wcel 1726   <.cop 3819   class class class wbr 4214   ` cfv 5456   NNcn 10002   EEcee 25829    Btwn cbtwn 25830    Colinear ccolin 25973
This theorem is referenced by:  btwncolinear2  26006  btwncolinear3  26007  btwncolinear4  26008  btwncolinear5  26009  btwncolinear6  26010  idinside  26020  btwnconn1lem12  26034  brsegle2  26045  broutsideof2  26058  outsidele  26068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-iota 5420  df-fv 5464  df-oprab 6087  df-colinear 25977
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