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Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsgplpte21c 25301 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 25302 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 25303 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 25304 The predicate "is a degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              Ibg

Theoremsgplpte21d1 25305 The extremities belong to a segment. (For my private use only. Don't use.) (Contributed by FL, 29-Jul-2016.)
PPoints              Ibg

Theoremsgplpte21d2 25306 The extremities belong to a segment. (For my private use only. Don't use.) (Contributed by FL, 29-Jul-2016.)
PPoints              Ibg

Theoremsegline 25307 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 25308 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 25309 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 25310* 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 25311 The segments xy and yx are equal. (For my private use only. Don't use.) (Contributed by FL, 17-Jul-2016.)
PPoints              Ibg

Theorembsstrs 25312 Being on the same side is a transitive relation. Segment version of bsstr 25294. (For my private use only. Don't use.) (Contributed by FL, 14-Jul-2016.)
PPoints       PLines              Ibg

Theoremnbssntrs 25313 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 25314 Extend class notation with the non-degenerated segment symbol.
Segments

Definitiondf-Seg 25315 The non-degenerated segments. (For my private use only. Don't use.) (Contributed by FL, 7-Mar-2016.)
Segments Ibg PPoints PPoints

Theoremnds 25316 The non-degenerated segments. (For my private use only. Don't use.) (Contributed by FL, 7-Mar-2016.)
PPoints              Ibg Segments

Syntaxcray2 25317 Extend class notation with the ray symbol.
ray

Definitiondf-ray2 25318* 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 25319 A degenerated ray. (For my private use only. Don't use.) (Contributed by FL, 13-Apr-2016.)
PPoints       ray       Ibg

Theoremisray 25320* A non-degenerated ray. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
PPoints              ray       Ibg                     btw

Theoremsegray 25321 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 25322 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 25323 Extend class notation with the angle symbol.
angle

Definitiondf-angle 25324* 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 25325 Extend class notation with the triangle symbol.
triangle

Definitiondf-triangle 25326* 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 25327 Extend class notation with triangle congruence.
trcng

Syntaxcsas 25328 Extend class notation with the same side relation.
ss

Definitiondf-sside 25329* 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 25330* 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 25331 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 25332 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 25333 "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 25334 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 25335 Halfplanes are distinct. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
PPoints       PLines       ss       Ibg

Syntaxchalfp 25336 Extend class notation with the Halfplane symbol.
Halfplane

Definitiondf-halfplane 25337* 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 25338 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 25339* 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 25340 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 25341 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 25342* 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 25343 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 25344 Extend class notation with the convex symbol.
convex

Definitiondf-cnvx 25345* 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 25346 Extend class notation with the segment congruence selector.
segc

Syntaxcibcg 25347 Extend class notation with the class of all planar, incidence, betweenness, congruence geometries.
Ibcg

Definitiondf-segc 25348 Segment congruence selector. (Contributed by FL, 1-Apr-2016.)
segc

Syntaxcangc 25349 Extend class notation with the angle congruence selector.
angc

Definitiondf-angc 25350 Angle congruence selector. (Contributed by FL, 1-Apr-2016.)
angc

Syntaxcangtrg 25351 Extend class notation a function returning an angle in a triangle.
angtrg

Definitiondf-angtrg 25352* Select an angle in a triangle. Definition 2 of [AitkenIBCG] p. 3. (Contributed by FL, 2-Sep-2016.)
angtrg Ig PPointsdWords angle

Syntaxcsegtrg 25353 Extend class notation a function returning a segment in a triangle.
segtrg

Definitiondf-segtrg 25354* Select a segment in a triangle. (Contributed by FL, 2-Sep-2016.)
segtrg Ig PPointsdWords angle

Definitiondf-trcng 25355* 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 25356* 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 25357* 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 25358 Extend class notation with the slices symbol.
slices

Definitiondf-slices 25359* 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 25360 Extend class notation with the cutpoint symbol.
Cut

Definitiondf-Cut 25361* 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 25362 Extend class notation with the neutral geometry symbol.
Neug

Definitiondf-neug 25363* 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 25364 Extend class notation with the Circle symbol.
Circle

Definitiondf-crcl 25365* 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

16.12  Mathbox for Jeff Hankins

16.12.1  Miscellany

Theorema1i13 25366 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)

Theorema1i4 25367 Add an antecedent to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)

Theorema1i14 25368 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)

Theorema1i24 25369 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)

Theorema1i34 25370 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)

Theoremimp5gOLD 25371 An importation inference. (Moved into main set.mm as imp5g 586 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremimp55OLD 25372 An importation inference. (Moved into main set.mm as imp55 587 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremimp511OLD 25373 An importation inference. (Moved into main set.mm as imp511 588 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremexp5d 25374 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp5g 25375 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp5j 25376 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp5k 25377 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp5l 25378 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp56 25379 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp58 25380 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp510 25381 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp511 25382 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp512 25383 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

TheoremmtordOLD 25384 A modus tollens deduction involving disjunction. (Moved into main set.mm as mtord 644 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 15-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3com12d 25385 Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.)

Theoremimp5p 25386 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)

Theoremimp5q 25387 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)

Theoremecase13d 25388 Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)

TheoremeqeuOLD 25389* A condition which implies existential uniqueness. (Moved into main set.mm as eqeu 2873 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 8-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsubtr 25390 Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Theoremsubtr2 25391 Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

TheoremcnvresimaOLD 25392 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) (Moved to cnvresima 5068 in main set.mm and may be deleted by mathbox owner, JGH. --NM 23-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtrer 25393* A relation intersected with its converse is an equivalence relation if the relation is transitive. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremelicc3 25394 An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.)

TheoremccidOLD 25395 A closed interval with identical lower and upper bounds is a singleton. (Moved into main set.mm as iccid 10579 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremioodisjOLD 25396 If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Moved into main set.mm as ioodisj 10643 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremfinminlem 25397* A useful lemma about finite sets. If a property holds for a finite set, it holds for a minimal set. (Contributed by Jeff Hankins, 4-Dec-2009.)

Theoremdivcan7OLD 25398 Cancel equal divisors in a division. (Contributed by Jeff Hankins, 29-Sep-2013.) (Moved to divcan7 9349 in main set.mm and may be deleted by mathbox owner, JGH. --NM 21-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoreminfleOLD 25399* If a nonempty set of real numbers has a lower bound, its infimum is less than or equal to any of its elements. (Contributed by Jeff Hankins, 15-Sep-2013.) (Moved to infmrlb 9615 in main set.mm and may be deleted by mathbox owner, JGH. --NM 21-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremgtinf 25400* Any number greater than an infimum is greater than some element of the set. (Contributed by Jeff Hankins, 29-Sep-2013.)

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