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Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisconcl5a 26101* 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.)
PLines       PPoints       Ig

Theoremisconcl5ab 26102* 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.)
PLines       PPoints       Ig

Theoremisconcl6a 26103* 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.)
PLines       PPoints       Ig

Theoremisconcl6ab 26104* 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.)
PLines       PPoints       Ig

Theoremisconcl7a 26105 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.)
PLines       Ig

Syntaxcbtw 26106 Extend class notation with the betweenness relation.
btw

Syntaxcibg 26107 Extend class notation with the class of all planar incidence betweenness geometries.
Ibg

Definitiondf-btw 26108 Definition of btw. (Contributed by FL, 1-Apr-2016.)
btw

Definitiondf-ibg2 26109* 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 Ig PPoints PLines coln btw

Theoremisibg2 26110* The predicate "is an incidence-betweenness geometry".

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

B-2 If and are distinct, then there are points , , such that is before and , is between and , is after and .

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

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

PPoints       PLines       btw       coln       Ibg Ig

Theoremisibg1a 26111 An incidence-betweenness geometry is an incidence geometry. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
Ibg       Ig

Theoremisibg2aa 26112* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
PPoints       PLines       btw       coln                                   Ibg

Theoremisibg2aalem1 26113* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)
PPoints       PLines

Theoremisibg2aalem2 26114* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)

Theoremisibg2aalem3 26115* Properties of an incidence-betweenness geometry. (Contributed by FL, 10-Aug-2016.)

Theoremisib2g1a1 26116 If is between and , it is between and (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1a2 26117 If is between and , then , , are collinear . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg                     coln

Theoremisibg2a 26118* 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.)
PPoints       btw       Ibg

Theoremisibg2a1 26119* Two distinct points , have a point before them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
PPoints       btw       Ibg

Theoremisibg2a2 26120* Two distinct points , have a point between them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
PPoints       btw       Ibg

Theoremisibg2a3 26121* Two distinct points , have a point after them. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
PPoints       btw       Ibg

Theoremisibg1a3a 26122 If is between and , then and , are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1spa 26123 If is between and , then and , are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1a5a 26124 If is between and , then and , are distinct . (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1a6 26125 If is between and , it belongs to the line XZ (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1a7 26126 If is between and , belongs to the line YZ (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theoremisibg1a8 26127 If is between and , belongs to the line XY (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints       btw       Ibg

Theorembsstr 26128 Being on the same side is a transitive relation. (For my private use only. Don't use.) (Contributed by FL, 10-Jul-2016.)
PPoints       PLines       btw       Ibg

Theoremnbssntr 26129 IF and are not on the same side, and and are not on the same side then and are on the same side. (For my private use only. Don't use.) (Contributed by FL, 10-Jul-2016.)
PPoints       PLines       btw       Ibg

Syntaxcseg 26130 Extend class notation with the segment symbol.

Definitiondf-seg2 26131* 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.)
Ibg PPoints PPoints PPoints btw

Theoremsgplpte21 26132* The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg              btw

Theoremsgplpte21a 26133* The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg              btw

Theoremsgplpte21b 26134 The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg              btw

Theoremsgplpte21c 26135 The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg              btw

Theoremsgplpte21d 26136 The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg              btw

Theoremsgplpte21e 26137 The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg              btw

Theoremsgplpte22 26138 The predicate "is a degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg

Theoremsgplpte21d1 26139 The extremities belong to a segment. (For my private use only. Don't use.) (Contributed by FL, 29-Jul-2016.)
PPoints              Ibg

Theoremsgplpte21d2 26140 The extremities belong to a segment. (For my private use only. Don't use.) (Contributed by FL, 29-Jul-2016.)
PPoints              Ibg

Theoremsegline 26141 A segment is a part of a line. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg

Theorempxysxy 26142 Points between and belong to the segment XY. (For my private use only. Don't use.) (Contributed by FL, 17-Jul-2016.)
PPoints              Ibg              btw

Theoremlppotoslem 26143 To show that a point doesn't belong to a line . (For my private use only. Don't use.) (Contributed by FL, 14-Jul-2016.)

Theoremlppotos 26144* Given a line and a point not on this line. There exists a point on the other side of the line. (For my private use only. Don't use.) (Contributed by FL, 10-Aug-2016.)
PPoints       PLines              Ibg

Theoremxsyysx 26145 The segments xy and yx are equal. (For my private use only. Don't use.) (Contributed by FL, 17-Jul-2016.)
PPoints              Ibg

Theorembsstrs 26146 Being on the same side is a transitive relation. Segment version of bsstr 26128. (For my private use only. Don't use.) (Contributed by FL, 14-Jul-2016.)
PPoints       PLines              Ibg

Theoremnbssntrs 26147 IF and are not on the same side, and and are not on the same side then and are on the same side. (For my private use only. Don't use.) (Contributed by FL, 14-Jul-2016.)
PPoints       PLines              Ibg

SyntaxcSeg 26148 Extend class notation with the non-degenerated segment symbol.
Segments

Definitiondf-Seg 26149 The non-degenerated segments. (For my private use only. Don't use.) (Contributed by FL, 7-Mar-2016.)
Segments Ibg PPoints PPoints

Theoremnds 26150 The non-degenerated segments. (For my private use only. Don't use.) (Contributed by FL, 7-Mar-2016.)
PPoints              Ibg Segments

Syntaxcray2 26151 Extend class notation with the ray symbol.
ray

Definitiondf-ray2 26152* Definition of the ray xy degenerated or not. Definition 11 of [AitkenIBG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
ray Ibg PPoints PPoints PPoints btw

Theoremisray2 26153 A degenerated ray. (For my private use only. Don't use.) (Contributed by FL, 13-Apr-2016.)
PPoints       ray       Ibg

Theoremisray 26154* A non-degenerated ray. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              ray       Ibg                     btw

Theoremsegray 26155 A segment is a part of a ray. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              ray       Ibg

Theoremrayline 26156 A ray is a part of a line. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              ray       Ibg

Syntaxcangle 26157 Extend class notation with the angle symbol.
angle

Definitiondf-angle 26158* Definition of an angle. Definition 17 of [AitkenIBG] p. 10. The angles can't be degenerated. Contrary to the concept of degenerated line or segment, the concept of degenerated angle no longer simplifies the wording. (For my private use only. Don't use.) (Contributed by FL, 7-Apr-2016.)
angle Ibg PPointsdWords ray ray

Syntaxctriangle 26159 Extend class notation with the triangle symbol.
triangle

Definitiondf-triangle 26160* Definition of a triangle, degenerated or not. Definition 23 of [AitkenIBG] p. 15. (For my private use only. Don't use.) (Contributed by FL, 7-Apr-2016.)
triangle Ibg PPointsdWords

Syntaxctrcng 26161 Extend class notation with triangle congruence.
trcng

Syntaxcsas 26162 Extend class notation with the same side relation.
ss

Definitiondf-sside 26163* Two points not on are on the same side of if the segment xy doesn't intersect . Definition 10 of [AitkenIBG] p. 4 (For my private use only. Don't use.) (Contributed by FL, 26-May-2016.)
ss Ibg PLines PPoints PPoints

Theoremisside0 26164* The predicate "Being on the same side of " (For my private use only. Don't use.) (Contributed by FL, 19-Jun-2016.)
PPoints       PLines       ss       Ibg

Theoremisside1 26165 The predicate "Being on the same side of " (For my private use only. Don't use.) (Contributed by FL, 19-Jun-2016.)
PPoints       PLines       ss       Ibg

Theoremisside 26166 The predicate "Being on the same side of " (For my private use only. Don't use.) (Contributed by FL, 19-Jun-2016.)
PPoints       PLines       ss       Ibg

Theorembosser 26167 "Being on the same side of " is an equivalence relation among points that are not on . (Contributed by FL, 10-Aug-2016.)
PPoints       PLines       ss       Ibg

Theorempdiveql 26168 The plane is divided into two equivalence classes by a line . (For my private use only. Don't use.) (Contributed by FL, 14-Jul-2016.)
PPoints       PLines       ss       Ibg

Theoremhpd 26169 Halfplanes are distinct. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
PPoints       PLines       ss       Ibg

Syntaxchalfp 26170 Extend class notation with the Halfplane symbol.
Halfplane

Definitiondf-halfplane 26171* Returns the halplanes delimited by . (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Halfplane Ibg PLines PPoints ss

Theoremaishp 26172 The half planes delimited by . (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Ibg              PPoints       ss       PLines       Halfplane

Theoremabhp 26173* The half planes delimited by . (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Ibg              PPoints       ss       PLines       Halfplane

Theoremabhp1 26174 The half planes delimited by . (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Ibg              PPoints       ss       PLines              Halfplane

Theoremabhp2 26175 The half planes delimited by . (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.) NEW
Ibg              PPoints       ss       PLines              Halfplane

Theorembhp2a 26176* The half planes delimited by . (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Ibg              PPoints       ss       PLines       Halfplane

Theorembhp3 26177 Every line divides the plane into exactly two half planes . (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Ibg              PPoints       ss       PLines       Halfplane

Syntaxcconvex 26178 Extend class notation with the convex symbol.
convex

Definitiondf-cnvx 26179* Definition of the convex subsets of points. Definition 24 of [AitkenIBG] p. 15. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
convex Ibg PPoints

Syntaxcsegc 26180 Extend class notation with the segment congruence selector.
segc

Syntaxcibcg 26181 Extend class notation with the class of all planar, incidence, betweenness, congruence geometries.
Ibcg

Definitiondf-segc 26182 Segment congruence selector. (Contributed by FL, 1-Apr-2016.)
segc

Syntaxcangc 26183 Extend class notation with the angle congruence selector.
angc

Definitiondf-angc 26184 Angle congruence selector. (Contributed by FL, 1-Apr-2016.)
angc

Syntaxcangtrg 26185 Extend class notation a function returning an angle in a triangle.
angtrg

Definitiondf-angtrg 26186* Select an angle in a triangle. Definition 2 of [AitkenIBCG] p. 3. (Contributed by FL, 2-Sep-2016.)
angtrg Ig PPointsdWords angle

Syntaxcsegtrg 26187 Extend class notation a function returning a segment in a triangle.
segtrg

Definitiondf-segtrg 26188* Select a segment in a triangle. (Contributed by FL, 2-Sep-2016.)
segtrg Ig PPointsdWords angle

Definitiondf-trcng 26189* Congruence of two triangles. Triangles are congruent if their sides and angles are congruent. "Technically, congruence is not a property of triangles themselves, but of triangles with a given ordering of their vertices. Triangles that are congruent under one ordering, might not be under other orderings." Definition 2 of [AitkenIBCG] p. 3. (Contributed by FL, 2-Sep-2016.)
trcng PPointsdWords PPointsdWords segtrgsegcsegtrg segtrgsegcsegtrg segtrgsegcsegtrg angtrgangcangtrg angtrgangcangtrg angtrgangcangtrg

Definitiondf-ibcg 26190* Incidence-Betweenness Geometry plus congruence axioms. (Here is an excerpt of Aitken's handout.)

Axiom (C-1). Segment congruence is an equivalence relation for line segments.

Axiom (C-2). Suppose that is a line segment and is a ray. Then there is a unique point on , distinct from , such that . The next axiom concerns copying dividing or intermediate points on a segment.

Axiom (C-3). Suppose that and are congruent line segments. If B is a point such that , then there is a point such that , , and .

Axiom (C-4). Angle congruence is an equivalence relation for angles.

Axiom (C-5). (Copying an angle) Suppose is an angle, and is a ray. Then on any given side of , there is a unique ray such that .

Axiom (C-6). (Copying a triangle) Suppose is a triangle, and is a segment such that . Then on any given side of , there is a point such that .

(For my private use only. Don't use.) Axiom C-1 C-2, C-3, C-4, C-5, C-6 of [AitkenIBCG] p. 2 . (Contributed by FL, 1-Apr-2016.)

Ibcg Ibg angc PPoints btw ray segc angle triangle dWords Halfplane Segments trcng

Theoremisibcg 26191* The predicate "is a incidence betwenness geometry with congruences". (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
angc       PPoints       Segments       btw       ray       segc                     Halfplane       triangle       angle       trcng       dWords       Ibcg Ibg

Syntaxcslices 26192 Extend class notation with the slices symbol.
slices

Definitiondf-slices 26193* Return the slices generated by the Dedekind cut of a set of points. Definition 1 of [AitkenNG] p. 2. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
slices Ibg PPoints convex convex

Syntaxccut 26194 Extend class notation with the cutpoint symbol.
Cut

Definitiondf-Cut 26195* Return the cutpoints of a set of points where and are the slices of a Dedekind cut of . Definition 2 of [AitkenNG] p. 2. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
Cut Ibg PPoints PPoints btw

Syntaxcneug 26196 Extend class notation with the neutral geometry symbol.
Neug

Definitiondf-neug 26197* Definition of a neutral geometry. Every Dedekind cut of a line has a cut point. (Axiom of Dedekind in [AitkenNG] p. 3.) (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
Neug Ibcg PLines slicesCut

Syntaxccircle 26198 Extend class notation with the Circle symbol.
Circle

Definitiondf-crcl 26199* Definition of a circle (degenerated or not). Definition 5 of [AitkenNG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
Circle Ibcg PPoints PPoints PPoints segc

18.14  Mathbox for Jeff Hankins

18.14.1  Miscellany

Theorema1i13 26200 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)

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