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Theorem colinrel 24752
Description: Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
colinrel  |-  Rel  Colinear

Proof of Theorem colinrel
Dummy variables  q  p  r  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5067 . 2  |-  Rel  `' { <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) }
2 df-colinear 24736 . . 3  |-  Colinear  =  `' { <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) }
32releqi 4788 . 2  |-  ( Rel  Colinear  <->  Rel  `' { <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) } )
41, 3mpbir 200 1  |-  Rel  Colinear
Colors of variables: wff set class
Syntax hints:    /\ wa 358    \/ w3o 933    /\ w3a 934    e. wcel 1696   E.wrex 2557   <.cop 3656   class class class wbr 4039   `'ccnv 4704   Rel wrel 4710   ` cfv 5271   {coprab 5875   NNcn 9762   EEcee 24588    Btwn cbtwn 24589    Colinear ccolin 24732
This theorem is referenced by:  brcolinear2  24753
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  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-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-colinear 24736
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