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Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcors 25201 Extend class notation with the function that returns two propositions joined with  \/.
 class  or s
 
Definitiondf-orc 25202 Function that returns two propositions joined with  \/. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  or s  =  ( x  e.  ( ( Kleene `  NN )  ^m  { 1 ,  2 } )  |->  ( (  \/ c  conc  ( x `  1 ) )  conc  ( x `  2 ) ) )
 
Syntaxcimpc 25203 Extend class notation with the function that returns two propositions joined with  ->.
 class  imp c
 
Definitiondf-impc 25204 Function that returns two propositions joined with  ->. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  imp c  =  ( x  e.  ( ( Kleene `  NN )  ^m  { 1 ,  2 } )  |->  ( (  => c  conc  ( x `  1 ) ) 
 conc  ( x `  2
 ) ) )
 
Syntaxcbic 25205 Extend class notation with the function that returns two propositions joined with 
<->.
 class  bi c
 
Definitiondf-bic 25206 Function that returns two propositions joined with  <->. Experimental. (Contributed by FL, 2-Feb-2014.)
 |-  bi c  =  ( x  e.  ( ( Kleene `  NN )  ^m  { 1 ,  2 } )  |->  ( (  <=> c  conc  ( x `
  1 ) ) 
 conc  ( x `  2
 ) ) )
 
Syntaxcprop 25207 Extend class notation to include the set of all propositioanl formulas.
 class  Prop
 
Definitiondf-prop 25208 The set of propositional formulas. Gallier p. 32. (Contributed by FL, 2-Feb-2014.)
 |-  Prop  =  ( ( ( P c " ( ZZ>= `  7 ) )  u. 
 { _|_ c } )  IndCls  ( { not c ,  and c ,  or s }  u.  { imp c ,  bi c } )
 )
 
Theoremsmbkle 25209 The symbols and variables of 
Prop belong to the Kleene star of  NN. (Contributed by FL, 2-Feb-2014.)
 |-  (
 ( P c "
 ( ZZ>= `  7 )
 )  u.  { _|_ c } )  C_  ( Kleene `
  NN )
 
Theoremintset 25210 The interval  ( 1 ... 2 ) in terms of its elements. (Contributed by FL, 2-Feb-2014.)
 |-  (
 1 ... 2 )  =  { 1 ,  2 }
 
Theoremfnckle 25211* The functions of  Prop. (Contributed by FL, 2-Feb-2014.)
 |-  A. f  e.  ( { not c ,  and c ,  or s }  u.  { imp c ,  bi c }
 ) E. n  e. 
 NN  f  e.  (
 ( Kleene `  NN )  ^m  ( ( Kleene `  NN )  ^m  ( 1 ... n ) ) )
 
Theoremfnckleb 25212 The functions of  Prop. (Contributed by FL, 2-Feb-2014.)
 |-  A. f  e.  ( { not c ,  and c ,  or s }  u.  { imp c ,  bi c }
 ) ( Fun  f  /\  ran  f  C_  ( Kleene `
  NN ) )
 
Theorempfsubkl 25213 Propositional formulas are a subset of the Kleene star of  NN. (Contributed by FL, 2-Feb-2014.)
 |-  Prop  C_  ( Kleene `  NN )
 
Theorempvp 25214 Propositional variables are propositions . (Contributed by FL, 2-Feb-2014.)
 |-  ( P c " ( ZZ>= `  7 ) )  C_  Prop
 
Theoremcndpv 25215 Condition to be a propositional variable. (Contributed by FL, 2-Feb-2014.)
 |-  (
 ( N  e.  ZZ  /\  7  <_  N )  ->  { <. 1 ,  N >. }  e.  ( P c " ( ZZ>= `  7 ) ) )
 
Theoremphpf 25216  ph c is a propositional formula. (Contributed by FL, 2-Feb-2014.)
 |-  ph c  e.  Prop
 
Theorempspf 25217  ps c is a propositional formula. (Contributed by FL, 2-Feb-2014.)
 |-  ps c  e.  Prop
 
Theorempgapspf 25218  ( ph  /\ 
ps ) is a propositional formula. We use variables. (Contributed by FL, 2-Feb-2014.)
 |-  (
 (  /\ c  conc  ph c
 )  conc  ps c )  e.  Prop
 
Theorempgapspf2 25219  ( ph  /\ 
ps ) is a propositional formula. Here we use meta-variables. (Contributed by FL, 2-Feb-2014.)
 |-  (
 ( P  e.  Prop  /\  Q  e.  Prop )  ->  ( (  /\ c  conc  P )  conc  Q )  e.  Prop )
 
Syntaxcderv 25220 Extend class notation with the relation "derives in one step".
 class  derv
 
Definitiondf-derv 25221* The relation  u derives  v in one step in the grammar  g. Experimental. (Contributed by FL, 15-Jul-2012.)
 |-  derv  =  ( g  e.  Grammar  |->  {
 <. x ,  y >.  |  ( x  e.  ( Kleene `
  g )  /\  y  e.  ( Kleene `  g )  /\  E. u  e.  ( Kleene `  g ) E. v  e.  ( Kleene `
  g ) E. p  e.  ( Kleene `  g ) E. q  e.  ( Kleene `  g )
 ( x  =  ( ( u (  conc  `  g ) p ) (  conc  `  g ) v )  /\  <. p ,  q >.  e.  ( prdct `  g )  /\  y  =  ( ( u (  conc  `  g
 ) q ) ( 
 conc  `  g ) v ) ) ) }
 )
 
16.11.61  Planar geometry
 
Syntaxcpoints 25222 Extend class notation with the class of all Points.
 class PPoints
 
Definitiondf-points 25223 Definition of PPoints. (Contributed by FL, 1-Apr-2016.)
 |- PPoints  =  Base
 
Syntaxcplines 25224 Extend class notation with the class of all Lines.
 class PLines
 
Definitiondf-plines 25225 Definition of PLines. (Contributed by FL, 1-Apr-2016.)
 |- PLines  = Scalar
 
Syntaxcig 25226 Extend class notation with the class of all planar incidence geometries.
 class Ig
 
Definitiondf-ig2 25227* Definition of a geometry that can build on the axioms of incidence. Definition of an Incidence-Betweenness Geometry in [AitkenIBG] p. 1-2. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
 |- Ig  =  { f  |  [. (PPoints `  f )  /  g ]. [. (PLines `  f
 )  /  h ]. ( A. l  e.  h  l  C_  g  /\  ( A. x  e.  g  A. y  e.  g  ( x  =/=  y  ->  E! l  e.  h  ( x  e.  l  /\  y  e.  l
 ) )  /\  A. l  e.  h  E. x  e.  g  E. y  e.  g  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l )  /\  E. x  e.  g  E. y  e.  g  E. z  e.  g  (
 ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z
 )  /\  A. l  e.  h  -.  ( x  e.  l  /\  y  e.  l  /\  z  e.  l ) ) ) ) }
 
Theorembisig0 25228* Definition of a geometry that can build on the axioms of incidence. Definition of an Incidence-Betweenness Geometry in [AitkenIBG] p. 1-2. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   =>    |-  ( I  e. Ig  <->  ( I  e. 
 _V  /\  ( A. l  e.  L  l  C_  P  /\  A. x  e.  P  A. y  e.  P  ( x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  e.  l ) )  /\  A. l  e.  L  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l ) )  /\  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. l  e.  L  -.  ( x  e.  l  /\  y  e.  l  /\  z  e.  l
 ) ) ) )
 
Theoremisig1a2 25229* A line is a set of points. This axiom is not needed. (Let's recall that the incidence relation can be formalized as an abstract relation. And that the belonging relationship is only an interpretation.) However Wayne Aitken adds this axiom to his system and I will follow him. The definitions below will take advantage of it. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   =>    |-  ( ph  ->  A. l  e.  L  l  C_  P )
 
Theoremisig12 25230 A line is a set of points. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  A  e.  L )   =>    |-  ( ph  ->  A  C_  P )
 
Theoremisig22 25231* There is only one line passing through two distinct points. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   =>    |-  ( ph  ->  A. x  e.  P  A. y  e.  P  ( x  =/=  y  ->  E! l  e.  L  ( x  e.  l  /\  y  e.  l
 ) ) )
 
Theoremisig2a2 25232* There is only one line passing through two distinct points. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  E! l  e.  L  ( A  e.  l  /\  B  e.  l )
 )
 
Theoremelhaltdp 25233* Every line has at least two distinct points. (For my private use only. Don't use.) (Contributed by FL, 28-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   =>    |-  ( ph  ->  A. l  e.  L  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  l  /\  y  e.  l
 ) )
 
Theoremelhaltdp2 25234* Every line has at least two distinct points. (For my private use only. Don't use.) (Contributed by FL, 28-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  A  e.  L )   =>    |-  ( ph  ->  E. x  e.  P  E. y  e.  P  ( x  =/=  y  /\  x  e.  A  /\  y  e.  A ) )
 
Theoremelhalop2 25235* Every line has at least one point. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  E. x  e.  P  x  e.  M )
 
Theoremtethpnc 25236* There exists three non colinear points. (For my private use only. Don't use.) (Contributed by FL, 28-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   =>    |-  ( ph  ->  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. l  e.  L  -.  ( x  e.  l  /\  y  e.  l  /\  z  e.  l
 ) ) )
 
Theoremtethpnc2 25237* There exists three non colinear points. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  -.  ( x  e.  M  /\  y  e.  M  /\  z  e.  M ) ) )
 
Theoremgltpntl 25238* Given a line, there exists a point not on this line. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  E. x  e.  P  x  e/  M )
 
Theoremgltpntl2 25239* Given a line, there exists a point not on this line. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  P  =  (PPoints `  I )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  I  e. Ig )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  E. x  x  e.  ( P  \  M ) )
 
Theoremaline 25240 A line is not empty. (Contributed by FL, 10-Aug-2016.)
 |-  ( ph  ->  I  e. Ig )   &    |-  L  =  (PLines `  I )   &    |-  ( ph  ->  M  e.  L )   =>    |-  ( ph  ->  M  =/= 
 (/) )
 
Theoremtpne 25241 The plane is not empty. Exercise 5 of [AitkenIBG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 29-Apr-2016.)
 |-  P  =  (PPoints `  I )   &    |-  ( ph  ->  I  e. Ig )   =>    |-  ( ph  ->  P  =/=  (/) )
 
Syntaxcline 25242 Extend class notation with the class of all lines.
 class  line
 
Definitiondf-li 25243* Definition of the line xy. It also defines a degenerate line. Definition 4 of [AitkenIBG] p. 2. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
 |-  line  =  ( f  e. Ig  |->  ( x  e.  (PPoints `  f
 ) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  ( iota_ l  e.  (PLines `  f
 ) ( x  e.  l  /\  y  e.  l ) ) ,  { x } )
 ) )
 
Theoremlinevala2 25244* Definition of the line xy. It also defines a degenerate line. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   =>    |-  ( ph  ->  M  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_
 l  e.  L ( x  e.  l  /\  y  e.  l )
 ) ,  { x } ) ) )
 
Theoremlineval222 25245* The line passing through two distinct points  A and 
B. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A M B )  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l
 ) ) )
 
Theoremlineval42 25246 Any line to which  A and  B are incident is the line  ( A M B ). (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  A  e.  N )   &    |-  ( ph  ->  B  e.  N )   &    |-  ( ph  ->  N  e.  L )   =>    |-  ( ph  ->  N  =  ( A M B ) )
 
Theoremlineval12 25247 The line passing through two distinct points  A and 
B is a line. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A M B )  e.  L )
 
Theoremlineval22 25248 The points  A and  B belong to the line passing through two distinct points  A and  B. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A  e.  ( A M B )  /\  B  e.  ( A M B ) ) )
 
Theoremlineval3a 25249 Value of a degenerate line. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   =>    |-  ( ph  ->  ( A M A )  =  { A } )
 
Theoremlineval12a 25250 The line passing through two distinct points  A and 
B is a set of points . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( A M B )  C_  P )
 
Theoremlineval2a 25251 The point  A belongs to the line passing through it . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  A  e.  ( A M B ) )
 
Theoremlineval2b 25252 The point  B belongs to the line passing through it . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  B  e.  ( A M B ) )
 
Theoremlineval4a 25253 The line AB is the line BA. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   =>    |-  ( ph  ->  ( A M B )  =  ( B M A ) )
 
Theoremlineval5a 25254 If  C is a point of AB, AB = CB. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  ( A M B ) )   &    |-  ( ph  ->  C  =/=  B )   =>    |-  ( ph  ->  ( A M B )  =  ( C M B ) )
 
Theoremlineval6a 25255 If  C is a point of AB, AB = AC (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  M  =  ( line `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  P )   &    |-  ( ph  ->  B  e.  P )   &    |-  ( ph  ->  C  e.  ( A M B ) )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  ( A M B )  =  ( A M C ) )
 
Syntaxccol 25256 Extend class notation with the class of all collinear points sets.
 class coln
 
Definitiondf-col 25257* Definition of collinear points. Definition 5 of [AitkenIBG] p. 3. (Contributed by FL, 1-Apr-2016.)
 |- coln  =  ( f  e. Ig  |->  { x  |  E. l  e.  (PLines `  f ) x  C_  l } )
 
Theoremiscola2 25258* The predicate "being collinear points". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  ( ph  ->  G  e. Ig )   =>    |-  ( ph  ->  (coln `  G )  =  { x  |  E. l  e.  (PLines `  G ) x  C_  l } )
 
Theoremiscol2 25259* The predicate "being collinear points". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  ( A  e.  (coln `  G ) 
 <-> 
 E. l  e.  (PLines `  G ) A  C_  l ) )
 
Theoremiscol3 25260* Collinear points are points. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  A  e.  (coln `  G ) )   &    |-  P  =  (PPoints `  G )   =>    |-  ( ph  ->  A. x  e.  A  x  e.  P )
 
Syntaxccon2 25261 Extend class notation with the class of all concurrent lines.
 class con
 
Definitiondf-con2 25262* Definition of concurrent lines. Definition 6 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |- con  =  ( f  e. Ig  |->  { x  e.  ~P (PLines `  f
 )  |  |^| x  =/= 
 (/) } )
 
Theoremisconcl1b 25263* The predicate "are concurrent lines". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  ( ph  ->  F  e. Ig )   =>    |-  ( ph  ->  (con `  F )  =  { x  e.  ~P (PLines `  F )  |  |^| x  =/=  (/) } )
 
Theoremisconcl2b 25264 The predicate "are concurrent lines". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  ( ph  ->  F  e. Ig )   =>    |-  ( ph  ->  ( A  e.  (con `  F )  <->  ( A  e.  ~P (PLines `  F )  /\  |^| A  =/=  (/) ) ) )
 
Theoremisconcl3b 25265 Only lines can be concurrent. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  ( ph  ->  F  e. Ig )   &    |-  ( ph  ->  A  e.  (con `  F ) )   =>    |-  ( ph  ->  A 
 C_  (PLines `  F )
 )
 
Theoremisconcl4b 25266 Concurrent lines have at least one point in common. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  ( ph  ->  F  e. Ig )   &    |-  ( ph  ->  A  e.  (con `  F ) )   =>    |-  ( ph  ->  |^|
 A  =/=  (/) )
 
Theoremisconcl5a 25267* Two distinct lines intersect in at most one point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  L  =  (PLines `  F )   &    |-  P  =  (PPoints `  F )   &    |-  ( ph  ->  F  e. Ig )   &    |-  ( ph  ->  L1  e.  L )   &    |-  ( ph  ->  L 2  e.  L )   &    |-  ( ph  ->  L1  =/=  L 2 )   =>    |-  ( ph  ->  E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
 
Theoremisconcl5ab 25268* Two distinct lines intersect in at most one point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  L  =  (PLines `  F )   &    |-  P  =  (PPoints `  F )   &    |-  ( ph  ->  F  e. Ig )   &    |-  ( ph  ->  L1  e.  L )   &    |-  ( ph  ->  L 2  e.  L )   &    |-  ( ph  ->  L1  =/=  L 2 )   =>    |-  ( ph  ->  E* p ( p  e.  L1  /\  p  e.  L 2 ) )
 
Theoremisconcl6a 25269* Two distinct non-parallel lines intersect in one and only point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  L  =  (PLines `  F )   &    |-  P  =  (PPoints `  F )   &    |-  ( ph  ->  F  e. Ig )   &    |-  ( ph  ->  L1  e.  L )   &    |-  ( ph  ->  L 2  e.  L )   &    |-  ( ph  ->  L1  =/=  L 2 )   &    |-  ( ph  ->  ( L1  i^i  L 2 )  =/=  (/) )   =>    |-  ( ph  ->  E! p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
 
Theoremisconcl6ab 25270* Two distinct non-parallel lines intersect in one and only point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  L  =  (PLines `  F )   &    |-  P  =  (PPoints `  F )   &    |-  ( ph  ->  F  e. Ig )   &    |-  ( ph  ->  L1  e.  L )   &    |-  ( ph  ->  L 2  e.  L )   &    |-  ( ph  ->  L1  =/=  L 2 )   &    |-  ( ph  ->  ( L1  i^i  L 2 )  =/=  (/) )   =>    |-  ( ph  ->  E! p ( p  e.  L1  /\  p  e.  L 2 ) )
 
Theoremisconcl7a 25271 Two distinct non-parallel lines intersect in one and only point. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
 |-  L  =  (PLines `  G )   &    |-  ( ph  ->  G  e. Ig )   &    |-  ( ph  ->  L1  e.  L )   &    |-  ( ph  ->  L 2  e.  L )   &    |-  ( ph  ->  L1  =/=  L 2 )   &    |-  ( ph  ->  X  e.  ( L1 
 i^i  L 2 ) )   =>    |-  ( ph  ->  ( L1  i^i  L 2 )  =  { X } )
 
Syntaxcbtw 25272 Extend class notation with the betweenness relation.
 class btw
 
Syntaxcibg 25273 Extend class notation with the class of all planar incidence betweenness geometries.
 class Ibg
 
Definitiondf-btw 25274 Definition of btw. (Contributed by FL, 1-Apr-2016.)
 |- btw  =  +g
 
Definitiondf-ibg2 25275* Definition of a geometry that can build on the axioms of incidence and betweenness. Axioms B-1, B-2, B-3, B-4 of [AitkenIBG] p. 3-4. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
 |- Ibg  =  {
 f  e. Ig  |  [. (PPoints `  f )  /  g ]. [. (PLines `  f
 )  /  h ]. [. (coln `  f )  /  d ]. [. (btw `  f
 )  /  e ]. A. p  e.  g  A. q  e.  g  (
 ( p  =/=  q  ->  E. a  e.  g  E. b  e.  g  E. c  e.  g  ( p  e.  (
 a e q ) 
 /\  b  e.  ( p e q ) 
 /\  q  e.  ( p e c ) ) )  /\  A. r  e.  g  (
 ( ( { p ,  q ,  r }  e.  d  /\  ( p  =/=  q  /\  p  =/=  r  /\  q  =/=  r ) )  ->  ( ( p  e.  ( q e r )  /\  q  e/  ( p e r ) 
 /\  r  e/  ( p e q ) )  \/  ( p 
 e/  ( q e r )  /\  q  e.  ( p e r )  /\  r  e/  ( p e q ) )  \/  ( p 
 e/  ( q e r )  /\  q  e/  ( p e r )  /\  r  e.  ( p e q ) ) ) ) 
 /\  ( q  e.  ( p e r )  ->  ( q  e.  ( r e p )  /\  { p ,  q ,  r }  e.  d  /\  ( p  =/=  q  /\  p  =/=  r  /\  q  =/=  r ) ) ) 
 /\  A. l  e.  h  ( ( p  e/  l  /\  q  e/  l  /\  r  e/  l ) 
 ->  ( ( ( ( ( p e q )  i^i  l )  =  (/)  /\  ( (
 q e r )  i^i  l )  =  (/) )  ->  ( ( p e r )  i^i  l )  =  (/) )  /\  ( ( ( ( p e q )  i^i  l
 )  =/=  (/)  /\  (
 ( q e r )  i^i  l )  =/=  (/) )  ->  (
 ( p e r )  i^i  l )  =  (/) ) ) ) ) ) }
 
Theoremisibg2 25276* The predicate "is an incidence-betweenness geometry".

B-1 If  q is between  p and  r then it is between  r and  p and  p,  q,  r are collinear and distinct.

B-2 If  p and  q are distinct, then there are points  a,  b,  c such that  a is before  p and  q,  b is between  p and  q,  c is after  p and  q.

B-3 If three points  p,  q,  r are collinear and distinct then exactly one of the followings occurs:  p is between  q and  r,  q is between  p and  r,  r is between  p and  q.

B-4 "Being on the same side" is a transitive relation. If  p and  q are not on the same side of  l and  q and  r are not on the same side of  l then  p and  r are on the same side of  l. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)

 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  B  =  (btw `  G )   &    |-  C  =  (coln `  G )   =>    |-  ( G  e. Ibg  <->  ( G  e. Ig  /\ 
 A. p  e.  P  A. q  e.  P  ( ( p  =/=  q  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( p  e.  (
 a B q ) 
 /\  b  e.  ( p B q )  /\  q  e.  ( p B c ) ) )  /\  A. r  e.  P  ( ( ( { p ,  q ,  r }  e.  C  /\  ( p  =/=  q  /\  p  =/=  r  /\  q  =/=  r
 ) )  ->  (
 ( p  e.  (
 q B r ) 
 /\  q  e/  ( p B r )  /\  r  e/  ( p B q ) )  \/  ( p  e/  (
 q B r ) 
 /\  q  e.  ( p B r )  /\  r  e/  ( p B q ) )  \/  ( p  e/  (
 q B r ) 
 /\  q  e/  ( p B r )  /\  r  e.  ( p B q ) ) ) )  /\  (
 ( q  e.  ( p B r )  ->  ( q  e.  (
 r B p ) 
 /\  { p ,  q ,  r }  e.  C  /\  ( p  =/=  q  /\  p  =/=  r  /\  q  =/=  r
 ) ) )  /\  A. l  e.  L  ( ( p  e/  l  /\  q  e/  l  /\  r  e/  l )  ->  ( ( ( ( ( p B q )  i^i  l )  =  (/)  /\  ( (
 q B r )  i^i  l )  =  (/) )  ->  ( ( p B r )  i^i  l )  =  (/) )  /\  ( ( ( ( p B q )  i^i  l
 )  =/=  (/)  /\  (
 ( q B r )  i^i  l )  =/=  (/) )  ->  (
 ( p B r )  i^i  l )  =  (/) ) ) ) ) ) ) ) )
 
Theoremisibg1a 25277 An incidence-betweenness geometry is an incidence geometry. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
 |-  ( ph  ->  G  e. Ibg )   =>    |-  ( ph  ->  G  e. Ig )
 
Theoremisibg2aa 25278* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  B  =  (btw `  G )   &    |-  C  =  (coln `  G )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  M  e.  L )   &    |-  ( ph  ->  G  e. Ibg )   =>    |-  ( ph  ->  (
 ( X  =/=  Y  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( X  e.  (
 a B Y ) 
 /\  b  e.  ( X B Y )  /\  Y  e.  ( X B c ) ) )  /\  ( ( ( { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) )  ->  (
 ( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X 
 e/  ( Y B Z )  /\  Y  e.  ( X B Z ) 
 /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z ) 
 /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B Z )  ->  ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) ) )  /\  ( ( X  e/  M  /\  Y  e/  M  /\  Z  e/  M )  ->  (
 ( ( ( ( X B Y )  i^i  M )  =  (/)  /\  ( ( Y B Z )  i^i 
 M )  =  (/) )  ->  ( ( X B Z )  i^i 
 M )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  M )  =/=  (/)  /\  ( ( Y B Z )  i^i 
 M )  =/=  (/) )  ->  ( ( X B Z )  i^i  M )  =  (/) ) ) ) ) ) ) )
 
Theoremisibg2aalem1 25279* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   =>    |-  (
 ( X  e.  P  /\  Y  e.  P ) 
 ->  ( A. x  e.  P  A. y  e.  P  ( ( x  =/=  y  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( x  e.  ( a B y )  /\  b  e.  ( x B y )  /\  y  e.  ( x B c ) ) )  /\  A. z  e.  P  ( ( ( { x ,  y ,  z }  e.  C  /\  ( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z ) )  ->  ( ( x  e.  ( y B z )  /\  y  e/  ( x B z ) 
 /\  z  e/  ( x B y ) )  \/  ( x  e/  ( y B z )  /\  y  e.  ( x B z )  /\  z  e/  ( x B y ) )  \/  ( x 
 e/  ( y B z )  /\  y  e/  ( x B z )  /\  z  e.  ( x B y ) ) ) ) 
 /\  ( ( y  e.  ( x B z )  ->  (
 y  e.  ( z B x )  /\  { x ,  y ,  z }  e.  C  /\  ( x  =/=  y  /\  x  =/=  z  /\  y  =/=  z
 ) ) )  /\  A. l  e.  L  ( ( x  e/  l  /\  y  e/  l  /\  z  e/  l )  ->  ( ( ( ( ( x B y )  i^i  l )  =  (/)  /\  ( (
 y B z )  i^i  l )  =  (/) )  ->  ( ( x B z )  i^i  l )  =  (/) )  /\  ( ( ( ( x B y )  i^i  l
 )  =/=  (/)  /\  (
 ( y B z )  i^i  l )  =/=  (/) )  ->  (
 ( x B z )  i^i  l )  =  (/) ) ) ) ) ) )  ->  ( ( X  =/=  Y 
 ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( X  e.  (
 a B Y ) 
 /\  b  e.  ( X B Y )  /\  Y  e.  ( X B c ) ) )  /\  A. z  e.  P  ( ( ( { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) 
 ->  ( ( X  e.  ( Y B z ) 
 /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) ) 
 /\  ( ( Y  e.  ( X B z )  ->  ( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) )  /\  A. l  e.  L  ( ( X 
 e/  l  /\  Y  e/  l  /\  z  e/  l )  ->  ( ( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l )  =  (/) )  ->  (
 ( X B z )  i^i  l )  =  (/) )  /\  (
 ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  (
 ( Y B z )  i^i  l )  =/=  (/) )  ->  (
 ( X B z )  i^i  l )  =  (/) ) ) ) ) ) ) ) )
 
Theoremisibg2aalem2 25280* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
 |-  ( Z  e.  P  ->  (
 A. z  e.  P  ( ( ( { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) 
 ->  ( ( X  e.  ( Y B z ) 
 /\  Y  e/  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e.  ( X B z )  /\  z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B z )  /\  Y  e/  ( X B z )  /\  z  e.  ( X B Y ) ) ) ) 
 /\  ( ( Y  e.  ( X B z )  ->  ( Y  e.  ( z B X )  /\  { X ,  Y ,  z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  z  /\  Y  =/=  z ) ) )  /\  A. l  e.  L  ( ( X 
 e/  l  /\  Y  e/  l  /\  z  e/  l )  ->  ( ( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B z )  i^i  l )  =  (/) )  ->  (
 ( X B z )  i^i  l )  =  (/) )  /\  (
 ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  (
 ( Y B z )  i^i  l )  =/=  (/) )  ->  (
 ( X B z )  i^i  l )  =  (/) ) ) ) ) )  ->  (
 ( ( { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) )  ->  (
 ( X  e.  ( Y B Z )  /\  Y  e/  ( X B Z )  /\  Z  e/  ( X B Y ) )  \/  ( X 
 e/  ( Y B Z )  /\  Y  e.  ( X B Z ) 
 /\  Z  e/  ( X B Y ) )  \/  ( X  e/  ( Y B Z ) 
 /\  Y  e/  ( X B Z )  /\  Z  e.  ( X B Y ) ) ) )  /\  ( ( Y  e.  ( X B Z )  ->  ( Y  e.  ( Z B X )  /\  { X ,  Y ,  Z }  e.  C  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) ) )  /\  A. l  e.  L  ( ( X 
 e/  l  /\  Y  e/  l  /\  Z  e/  l )  ->  ( ( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l )  =  (/) )  ->  (
 ( X B Z )  i^i  l )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) ) ) ) ) ) ) )
 
Theoremisibg2aalem3 25281* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
 |-  ( M  e.  L  ->  (
 A. l  e.  L  ( ( X  e/  l  /\  Y  e/  l  /\  Z  e/  l ) 
 ->  ( ( ( ( ( X B Y )  i^i  l )  =  (/)  /\  ( ( Y B Z )  i^i  l )  =  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  l )  =/=  (/)  /\  ( ( Y B Z )  i^i  l )  =/=  (/) )  ->  ( ( X B Z )  i^i  l )  =  (/) ) ) ) 
 ->  ( ( X  e/  M  /\  Y  e/  M  /\  Z  e/  M ) 
 ->  ( ( ( ( ( X B Y )  i^i  M )  =  (/)  /\  ( ( Y B Z )  i^i 
 M )  =  (/) )  ->  ( ( X B Z )  i^i 
 M )  =  (/) )  /\  ( ( ( ( X B Y )  i^i  M )  =/=  (/)  /\  ( ( Y B Z )  i^i 
 M )  =/=  (/) )  ->  ( ( X B Z )  i^i  M )  =  (/) ) ) ) ) )
 
Theoremisib2g1a1 25282 If  Y is between  X and  Z, it is between  Z and  X (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   =>    |-  ( ph  ->  Y  e.  ( Z B X ) )
 
Theoremisibg1a2 25283 If  Y is between  X and  Z, then  X,  Y,  Z are collinear . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  C  =  (coln `  G )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   &    |-  ( ph  ->  Z  e.  P )   =>    |-  ( ph  ->  { X ,  Y ,  Z }  e.  C )
 
Theoremisibg2a 25284* Two distinct points have a point before, between and after them. (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  ( X  e.  ( a B Y )  /\  b  e.  ( X B Y )  /\  Y  e.  ( X B c ) ) )
 
Theoremisibg2a1 25285* Two distinct points  X,  Y have a point before them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  E. a  e.  P  X  e.  ( a B Y ) )
 
Theoremisibg2a2 25286* Two distinct points  X,  Y have a point between them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  E. b  e.  P  b  e.  ( X B Y ) )
 
Theoremisibg2a3 25287* Two distinct points  X,  Y have a point after them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  E. c  e.  P  Y  e.  ( X B c ) )
 
Theoremisibg1a3a 25288 If  Y is between  X and  Z, then  X and 
Y, are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   =>    |-  ( ph  ->  X  =/=  Y )
 
Theoremisibg1a4a 25289 If  Y is between  X and  Z, then  X and 
Z, are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   =>    |-  ( ph  ->  X  =/=  Z )
 
Theoremisibg1a5a 25290 If  Y is between  X and  Z, then  Y and 
Z, are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   =>    |-  ( ph  ->  Y  =/=  Z )
 
Theoremisibg1a6 25291 If  Y is between  X and  Z, it belongs to the line XZ (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   &    |-  M  =  ( line `  G )   =>    |-  ( ph  ->  Y  e.  ( X M Z ) )
 
Theoremisibg1a7 25292 If  Y is between  X and  Z,  X belongs to the line YZ (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   &    |-  M  =  ( line `  G )   =>    |-  ( ph  ->  X  e.  ( Y M Z ) )
 
Theoremisibg1a8 25293 If  Y is between  X and  Z,  Z belongs to the line XY (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  Z  e.  P )   &    |-  ( ph  ->  Y  e.  ( X B Z ) )   &    |-  M  =  ( line `  G )   =>    |-  ( ph  ->  Z  e.  ( X M Y ) )
 
Theorembsstr 25294 Being on the same side is a transitive relation. (For my private use only. Don't use.) (Contributed by FL, 10-Jul-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  ( ph  ->  X  e.  ( P  \  M ) )   &    |-  ( ph  ->  Y  e.  ( P  \  M ) )   &    |-  ( ph  ->  Z  e.  ( P  \  M ) )   &    |-  ( ph  ->  ( ( X B Y )  i^i 
 M )  =  (/) )   &    |-  ( ph  ->  (
 ( Y B Z )  i^i  M )  =  (/) )   =>    |-  ( ph  ->  (
 ( X B Z )  i^i  M )  =  (/) )
 
Theoremnbssntr 25295 IF  X and  Y are not on the same side, and  Y and  Z are not on the same side then 
X and  Z are on the same side. (For my private use only. Don't use.) (Contributed by FL, 10-Jul-2016.)
 |-  P  =  (PPoints `  G )   &    |-  L  =  (PLines `  G )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  M  e.  L )   &    |-  ( ph  ->  X  e.  ( P  \  M ) )   &    |-  ( ph  ->  Y  e.  ( P  \  M ) )   &    |-  ( ph  ->  Z  e.  ( P  \  M ) )   &    |-  ( ph  ->  ( ( X B Y )  i^i 
 M )  =/=  (/) )   &    |-  ( ph  ->  ( ( Y B Z )  i^i 
 M )  =/=  (/) )   =>    |-  ( ph  ->  ( ( X B Z )  i^i  M )  =  (/) )
 
Syntaxcseg 25296 Extend class notation with the segment symbol.
 class  seg
 
Definitiondf-seg2 25297* Definition of the segment xy degenerated or not. Definition 8 of [AitkenIBG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 28-Feb-2016.)
 |-  seg  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f
 ) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f )  |  ( z  e.  ( x (btw `  f )
 y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
 
Theoremsgplpte21 25298* The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  ( X S Y )  =  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } )
 
Theoremsgplpte21a 25299* The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   =>    |-  ( ph  ->  A. z
 ( z  e.  ( X S Y )  <->  ( z  e.  P  /\  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) ) ) )
 
Theoremsgplpte21b 25300 The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
 |-  P  =  (PPoints `  G )   &    |-  S  =  ( seg `  G )   &    |-  ( ph  ->  G  e. Ibg )   &    |-  ( ph  ->  X  e.  P )   &    |-  B  =  (btw `  G )   &    |-  ( ph  ->  Y  e.  P )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Z  e.  _V )   =>    |-  ( ph  ->  ( Z  e.  ( X S Y )  <->  ( Z  e.  P  /\  ( Z  e.  ( X B Y )  \/  Z  =  X  \/  Z  =  Y ) ) ) )
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