Theorem colinrel 32164

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.

Step	Hyp	Ref	Expression
1		relcnv 5503	. 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 32146	. . 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 >. ) ) }
3	2	releqi 5202	. 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 >. ) ) } )
4	1, 3	mpbir 221	1 |- Rel Colinear

Syntax hints:   /\ wa 384   \/ w3o 1036   /\ w3a 1037   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653   `'ccnv 5113   Rel wrel 5119   ` cfv 5888   {coprab 6651   NNcn 11020   EEcee 25768   Btwn cbtwn 25769   Colinear ccolin 32144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-colinear 32146

This theorem is referenced by:  brcolinear2  32165