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Theorem colinrel 25993
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 5244 . 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 25977 . . 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 4962 . 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 202 1  |-  Rel  Colinear
Colors of variables: wff set class
Syntax hints:    /\ wa 360    \/ w3o 936    /\ w3a 937    e. wcel 1726   E.wrex 2708   <.cop 3819   class class class wbr 4214   `'ccnv 4879   Rel wrel 4885   ` cfv 5456   {coprab 6084   NNcn 10002   EEcee 25829    Btwn cbtwn 25830    Colinear ccolin 25973
This theorem is referenced by:  brcolinear2  25994
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-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  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-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-colinear 25977
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