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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnaim12i 24001 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremnabi1 24002 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)

Theoremnabi2 24003 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)

Theoremnabi1i 24004 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremnabi2i 24005 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremnabi12i 24006 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)

Syntaxw3nand 24007 The double nand.

Definitiondf-3nand 24008 The double nand. This definition allows us to express the input of three variables only being false if all three are true. (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremdf3nandALT1 24009 The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremdf3nandALT2 24010 The double nand expressed in terms of negation and and. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremandnand1 24011 Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremimnand2 24012 An nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)

16.7.2  Predicate Calculus

Theoremquantriv 24013* Any wff can be trivially quantified, so long as the quantifier's set is distinct from said wff.

Theoremallt 24014 For all sets, is true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremalnof 24015 For all sets, is not true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremnalf 24016 Not all sets hold as true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremextt 24017 There exists a set that holds as true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremnextnt 24018 There does not exist a set, such that is not true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremnextf 24019 There does not exist a set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremunnf 24020 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremunnt 24021 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremmont 24022 There does not exist at most one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremmof 24023 There exist at most one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)

16.7.3  Misc. Single Axiom Systems

Theoremmeran1 24024 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)

Theoremmeran2 24025 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)

Theoremmeran3 24026 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)

Theoremwaj-ax 24027 A single axiom for propositional calculus offered by Wajsberg. (Contributed by Anthony Hart, 13-Aug-2011.)

Theoremlukshef-ax2 24028 A single axiom for propositional calculus offered by Lukasiewicz. (Contributed by Anthony Hart, 14-Aug-2011.)

Theoremarg-ax 24029 ? (Contributed by Anthony Hart, 14-Aug-2011.)

16.7.4  Connective Symmetry

Theoremnegsym1 24030 In the paper "On Variable Functors of Propositional Arguments", Lukasiewicz introduced a system that can handle variable connectives. This was done by introducing a variable, marked with a lowercase delta, which takes a wff as input. In the system, "delta " means that "something is true of ." "delta " can be substituted with , , , etc.

Later on, Meredith discovered a single axiom, in the form of delta delta delta . This represents the shortest theorem in the extended propositional calculus that cannot be derived as an instance of a theorem in propositional calculus.

A symmetry with . (Contributed by Anthony Hart, 4-Sep-2011.)

Theoremimsym1 24031 A symmetry with .

Theorembisym1 24032 A symmetry with .

Theoremconsym1 24033 A symmetry with .

Theoremdissym1 24034 A symmetry with .

Theoremnandsym1 24035 A symmetry with .

Theoremunisym1 24036 A symmetry with .

See negsym1 24030 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Theoremexisym1 24037 A symmetry with .

Theoremunqsym1 24038 A symmetry with .

Theoremamosym1 24039 A symmetry with .

Theoremsubsym1 24040 A symmetry with .

16.8  Mathbox for Chen-Pang He

16.8.1  Ordinal topology

Theoremontopbas 24041 An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)

Theoremonsstopbas 24042 The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.)

Theoremonpsstopbas 24043 The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.)

Theoremontgval 24044 The topology generated from an ordinal number is . (Contributed by Chen-Pang He, 10-Oct-2015.)

Theoremontgsucval 24045 The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.)

Theoremonsuctop 24046 A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)

Theoremonsuctopon 24047 One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.)
TopOn

Theoremordtoplem 24048 Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)

Theoremordtop 24049 An ordinal is a topology iff it is not its supermum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.)

Theoremonsucconi 24050 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)

Theoremonsuccon 24051 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)

Theoremordtopcon 24052 An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)

Theoremonintopsscon 24053 An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)

Theoremonsuct0 24054 A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)

Theoremordtopt0 24055 An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)

Theoremonsucsuccmpi 24056 The successor of a successor ordinal number is a compact topology, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 18-Oct-2015.)

Theoremonsucsuccmp 24057 The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.)

Theoremlimsucncmpi 24058 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)

Theoremlimsucncmp 24059 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)

Theoremordcmp 24060 Iff an ordinal topology is compact, the underlying set is its supermum (union) only when the ordinal is . (Contributed by Chen-Pang He, 1-Nov-2015.)

Theoremssoninhaus 24061 The ordinal topologies and are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)

Theoremonint1 24062 The ordinal T1 spaces are and , proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.)

Theoremoninhaus 24063 The ordinal Hausdorff spaces are and . (Contributed by Chen-Pang He, 10-Nov-2015.)

16.9  Mathbox for Jeff Hoffman

16.9.1  Inferences for finite induction on generic function values

Theoremfindabrcl 24067* Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)

16.9.2  gdc.mm

Theoremnnssi2 24068 Convert a theorem for real/complex numbers into one for natural numbers. (Contributed by Jeff Hoffman, 17-Jun-2008.)

Theoremnnssi3 24069 Convert a theorem for real/complex numbers into one for natural numbers. (Contributed by Jeff Hoffman, 17-Jun-2008.)

Theoremnndivlub 24071 A factor of a natural number cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.)

SyntaxcgcdOLD 24072 Extend class notation to include the gdc function.

Definitiondf-gcdOLD 24073* is the largest natural number that evenly divides both and . (Contributed by Jeff Hoffman, 17-Jun-2008.)

Theoremee7.2aOLD 24074 Lemma for Euclid's Elements, Book 7, proposition 2. The original mentions the smaller measure being 'continually subtracted' from the larger. Many authors interpret this phrase as mod . Here, just one subtraction step is proved to preserve the . The function will be used in other proofs for iterated subtraction. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

16.10  Mathbox for Wolf Lammen

Most of the theorems in the section "Logical implication" are about handling chains of implications: . With respect to chains, an rich set of rules clarify

- how to swap antecedents (com12, ...);

- how to drop antecedents (ax-mp, pm2.43, ...);

- how to add antecedents (a1i, ...)

- how to replace an antecedent (syl, ...);

- how to replace a consequent (ax-mp, syl, ...);

- what is, when an antecedent equals the consequent (ax-1, id, ...).

In all these cases, the operands of the chain have no inner structure, or it is of no importance. These chains are called "simple" here.

There is less support, when the operands are structured themselves. Some kinds of inner structure involving the operator are best handled by the symmetric operators and . But a nested, simple chain has no such convenient replacement. I can focus on antecedents here, since a consequent representing a chain is, in conjunction with its antecedents, just an extended simple chain again.

The following theorems show, how operations on nested chains appear somehow mirrored: The minor premises of the syllogisms look reverted, in comparison to their normal counterparts, and while adding an antecedent to a chain via a1i 12 is easy, in nested chains they can be easily dropped.

Theoremwl-jarri 24075 Dropping a nested antecedent. This theorem is one of two reversions of ja 155. Since ja 155 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

ax46 1746 is an instance of this idea.

Theoremwl-jarli 24076 Dropping a nested consequent. This theorem is one of two reversions of ja 155. Since ja 155 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

ax46 1746 is an instance of this idea.

Theoremwl-mps 24077 Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-syls1 24078 Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-syls2 24079 Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-adnestant 24080 A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid (see wl-adnestantALT 24081) (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-adnestantALT 24081 Proof of wl-adnestant 24080 not based on ax-3 9. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-adnestantd 24082 Deduction version of wl-adnestant 24080. Generalization of a2i 14, imim12i 55, imim1i 56 and imim2i 15, which can be proved by specializing its hypotheses, and some trivial rearrangements. This theorem clarifies in a more general way, under what conditions a wff may be introduced as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid (see wl-adnestantdALTOLD 24083). (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-adnestantdALTOLD 24083 Proof of wl-adnestantd 24082 not based on ax-3 9. (Contributed by Wolf Lammen, 4-Oct-2013.) (Moved to embantd 52 in main set.mm and may be deleted by mathbox owner, WL. --NM 14-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bitr1 24084 Closed form of bitri 242. Place before bitri 242. [ +33] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bitri 24085 An inference from transitive law for logical equivalence. [ -5] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bitrd 24086 Deduction form of bitri 242. [ -7] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bibi1 24087 Theorem *4.86 of [WhiteheadRussell] p. 122. Place this (and the following theorems) after bitr1. [ +22] (Contributed by NM, 3-Jan-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bibi1i 24088 Inference adding a biconditional to the right in an equivalence. Move after bibi1. [ -8] (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bibi1d 24089 Deduction adding a biconditional to the right in an equivalence. Move after bibi1i. [ -9] (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bibi2d 24090 Deduction adding a biconditional to the left in an equivalence. Move after bibi1d. [ -25] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-pm5.74lem 24091 Moving a common antecedent on one side of an equivalence. Place before pm5.74 237. [ +25] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-pm5.74 24092 Distribution of implication over biconditional. Theorem *5.74 of [ WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.) Replace and move biimt 327.. albi 1552 before it. [ -22] (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-pm5.32 24093 Distribution of implication over biconditional. Theorem *5.32 of [ WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Oct-2013.) Replace. [ -43] (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bitr 24094 Theorem *4.22 of [WhiteheadRussell] p. 117. Replace. [ -4] (Contributed by NM, 3-Jan-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-pm2.86i 24095 Inference based on pm2.86 96. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-dedlem0a 24096 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

16.11  Mathbox for Frédéric Liné

In the sequel "JFM" is the "Journal of Formalized Mathematics". http://mizar.uwb.edu.pl/JFM/mmlident.html

"CAT1"; means Bylinski Czeslaw, Introduction to Categories and Functors, Journal of Formalized Mathematics, 1990, volume 1, no 2, pages 409--420

"CAT2"; means Bylinski Czeslaw, Subcategories and Products of Categories, Journal of Formalized Mathematics, 1990, volume 1, no 4, pages 725--732

"CLASSES1" means Grzegorz Bancerek, Tarski's Classes and Ranks, Journal of Formalized Mathematics, 1990, volume 1, no 3, pages 563--567

"CLASSES2" means Bogdan Nowak and Grzegorz Bancerek, Universal Classes, Journal of Formalized Mathematics, may-august 1990, volume 1, nb 3, pages 595--600

"Bourbaki" means Bourbaki's treatise. The book General Topology is called TG (for Topologie Générale). The book Set Theory is called E (for théorie des Ensembles).

The treatise is translated in English.

More precisely, here are two examples of references:

"Bourbaki E II.32" means Set Theory, chapter II, 32nd page, "Bourbaki TG III.1" means General Topology, chapter III, 1st page.

The references are given according to the French edition.

"Viro" means O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov, Elementary topology. Available on the net.

"Goldblatt" means Robert Goldblatt, Topoi, the categorial analysis of logic, revised edition, Dover publications, Mineola, New-York, 2006

"Gallier" means Jean H. Gallier, "Logic For Computer Science -- Foundations of Automatic Theorem Proving". A new edition must be published in 2014 at Dover.

"Harju" means Tero Harju, "Lecture Notes on SemiGroups", unpublished, 1996. Available on the net.

In the following notices "experimental" means I have not yet sufficiently used a definition to be sure it is correct. Anyway I'm not the owner of the definition and you can use it as you wish if you think it is correct or replace it by a definition of your own if you think it is not.

16.11.1  Theorems from other workspaces

Theoremtpssg 24097 A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Revised by FL, 17-May-2016.)

16.11.2  Propositional and predicate calculus

Theoremneleq12d 24098 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)

Theoremr19.26-2a 24099 Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 20-May-2016.)

Theoremreubidvag 24100* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by FL, 17-Nov-2014.)

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