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Theorem btwncolinear5 26012
Description: Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
btwncolinear5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  C  Colinear  <. A ,  B >. ) )

Proof of Theorem btwncolinear5
StepHypRef Expression
1 btwncolinear1 26008 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  A  Colinear  <. B ,  C >. ) )
2 colinearperm4 26004 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  C  Colinear  <. A ,  B >. ) )
31, 2sylibd 207 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  C  Colinear  <. A ,  B >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1726   <.cop 3819   class class class wbr 4215   ` cfv 5457   NNcn 10005   EEcee 25832    Btwn cbtwn 25833    Colinear ccolin 25976
This theorem is referenced by:  btwnconn1lem12  26037  lineunray  26086  lineelsb2  26087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pr 4406
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 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-iota 5421  df-fv 5465  df-oprab 6088  df-colinear 25980
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