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

Theorembpolysum 26101* A sum for Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 22-May-2014.)
BernPoly

Theorembpolydiflem 26102* Lemma for bpolydif 26103. (Contributed by Scott Fenton, 12-Jun-2014.)
BernPoly BernPoly        BernPoly BernPoly

Theorembpolydif 26103 Calculate the difference between successive values of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 26-May-2014.)
BernPoly BernPoly

Theoremfsumkthpow 26104* A closed-form expression for the sum of -th powers. (Contributed by Scott Fenton, 16-May-2014.) (Revised by Mario Carneiro, 16-Jun-2014.)
BernPoly BernPoly

Theorembpoly2 26105 The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)
BernPoly

Theorembpoly3 26106 The Bernoulli polynomials at three. (Contributed by Scott Fenton, 8-Jul-2015.)
BernPoly

Theorembpoly4 26107 The Bernoulli polynomials at four. (Contributed by Scott Fenton, 8-Jul-2015.)
BernPoly ;

Theoremfsumcube 26108* Express the sum of cubes in closed terms. (Contributed by Scott Fenton, 16-Jun-2015.)

19.7.51  Rank theorems

Theoremrankung 26109 The rank of the union of two sets. Closed form of rankun 7784. (Contributed by Scott Fenton, 15-Jul-2015.)

Theoremranksng 26110 The rank of a singleton. Closed form of ranksn 7782. (Contributed by Scott Fenton, 15-Jul-2015.)

Theoremrankelg 26111 The membership relation is inherited by the rank function. Closed form of rankel 7767. (Contributed by Scott Fenton, 16-Jul-2015.)

Theoremrankpwg 26112 The rank of a power set. Closed form of rankpw 7771. (Contributed by Scott Fenton, 16-Jul-2015.)

Theoremrank0 26113 The rank of the empty set is (Contributed by Scott Fenton, 17-Jul-2015.)

Theoremrankeq1o 26114 The only set with rank is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)

19.7.52  Hereditarily Finite Sets

Syntaxchf 26115 The constant Hf is a class.
Hf

Definitiondf-hf 26116 Define the hereditarily finite sets. These are the finite sets whose elements are finite, and so forth. (Contributed by Scott Fenton, 9-Jul-2015.)
Hf

Theoremelhf 26117* Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.)
Hf

Theoremelhf2 26118 Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
Hf

Theoremelhf2g 26119 Hereditarily finiteness via rank. Closed form of elhf2 26118. (Contributed by Scott Fenton, 15-Jul-2015.)
Hf

Theorem0hf 26120 The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.)
Hf

Theoremhfun 26121 The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
Hf Hf Hf

Theoremhfsn 26122 The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
Hf Hf

Theoremhfadj 26123 Adjoining one HF element to an HF set preserves HF status. (Contributed by Scott Fenton, 15-Jul-2015.)
Hf Hf Hf

Theoremhfelhf 26124 Any member of an HF set is itself an HF set. (Contributed by Scott Fenton, 16-Jul-2015.)
Hf Hf

Theoremhftr 26125 The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.)
Hf

Theoremhfext 26126* Extensionality for HF sets depends only on comparison of HF elements. (Contributed by Scott Fenton, 16-Jul-2015.)
Hf Hf Hf

Theoremhfuni 26127 The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
Hf Hf

Theoremhfpw 26128 The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
Hf Hf

Theoremhfninf 26129 is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
Hf

19.8  Mathbox for Anthony Hart

19.8.1  Propositional Calculus

Theoremtb-ax1 26130 The first of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtb-ax2 26131 The second of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtb-ax3 26132 The third of three axioms in the Tarski-Bernays axiom system.

This axiom, along with ax-mp 8, tb-ax1 26130, and tb-ax2 26131, can be used to derive any theorem or rule that uses only . (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtbsyl 26133 The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremre1ax2lem 26134 Lemma for re1ax2 26135. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremre1ax2 26135 ax-2 6 rederived from the Tarski-Bernays axiom system. Often tb-ax1 26130 is replaced with this theorem to make a "standard" system. This is because this theorem is easier to work with, despite it being longer. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnaim1 26136 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)

Theoremnaim2 26137 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)

Theoremnaim1i 26138 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremnaim2i 26139 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremnaim12i 26140 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremnabi1 26141 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)

Theoremnabi2 26142 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)

Theoremnabi1i 26143 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremnabi2i 26144 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremnabi12i 26145 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)

Syntaxw3nand 26146 The double nand.

Definitiondf-3nand 26147 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 26148 The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremdf3nandALT2 26149 The double nand expressed in terms of negation and and. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremandnand1 26150 Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)

Theoremimnand2 26151 An nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)

19.8.2  Predicate Calculus

Theoremquantriv 26152* Any wff can be trivially quantified, so long as the quantifier's set is distinct from said wff.

Theoremallt 26153 For all sets, is true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremalnof 26154 For all sets, is not true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremnalf 26155 Not all sets hold as true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremextt 26156 There exists a set that holds as true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremnextnt 26157 There does not exist a set, such that is not true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremnextf 26158 There does not exist a set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremunnf 26159 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremunnt 26160 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremmont 26161 There does not exist at most one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)

Theoremmof 26162 There exist at most one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)

19.8.3  Misc. Single Axiom Systems

Theoremmeran1 26163 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)

Theoremmeran2 26164 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)

Theoremmeran3 26165 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)

Theoremwaj-ax 26166 A single axiom for propositional calculus offered by Wajsberg. (Contributed by Anthony Hart, 13-Aug-2011.)

Theoremlukshef-ax2 26167 A single axiom for propositional calculus offered by Lukasiewicz. (Contributed by Anthony Hart, 14-Aug-2011.)

Theoremarg-ax 26168 ? (Contributed by Anthony Hart, 14-Aug-2011.)

19.8.4  Connective Symmetry

Theoremnegsym1 26169 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 26170 A symmetry with .

Theorembisym1 26171 A symmetry with .

Theoremconsym1 26172 A symmetry with .

Theoremdissym1 26173 A symmetry with .

Theoremnandsym1 26174 A symmetry with .

Theoremunisym1 26175 A symmetry with .

See negsym1 26169 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Theoremexisym1 26176 A symmetry with .

Theoremunqsym1 26177 A symmetry with .

Theoremamosym1 26178 A symmetry with .

Theoremsubsym1 26179 A symmetry with .

19.9  Mathbox for Chen-Pang He

19.9.1  Ordinal topology

Theoremontopbas 26180 An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)

Theoremonsstopbas 26181 The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.)

Theoremonpsstopbas 26182 The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.)

Theoremontgval 26183 The topology generated from an ordinal number is . (Contributed by Chen-Pang He, 10-Oct-2015.)

Theoremontgsucval 26184 The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.)

Theoremonsuctop 26185 A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)

Theoremonsuctopon 26186 One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.)
TopOn

Theoremordtoplem 26187 Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)

Theoremordtop 26188 An ordinal is a topology iff it is not its supremum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.)

Theoremonsucconi 26189 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)

Theoremonsuccon 26190 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)

Theoremordtopcon 26191 An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)

Theoremonintopsscon 26192 An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)

Theoremonsuct0 26193 A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)

Theoremordtopt0 26194 An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)

Theoremonsucsuccmpi 26195 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 26196 The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.)

Theoremlimsucncmpi 26197 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)

Theoremlimsucncmp 26198 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)

Theoremordcmp 26199 An ordinal topology is compact iff the underlying set is its supremum (union) only when the ordinal is . (Contributed by Chen-Pang He, 1-Nov-2015.)

Theoremssoninhaus 26200 The ordinal topologies and are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)

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