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Theorem List for Metamath Proof Explorer - 23101-23200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremballotlemfrc 23101* Express the value of in terms of the newly defined . (Contributed by Thierry Arnoux, 21-Apr-2017.)

Theoremballotlemfrci 23102* Reverse counting preserves a tie at the first tie. (Contributed by Thierry Arnoux, 21-Apr-2017.)

Theoremballotlemfrceq 23103* Value of for a reverse counting . (Contributed by Thierry Arnoux, 27-Apr-2017.)

Theoremballotlemfrcn0 23104* Value of for a reversed counting , before the first tie, cannot be zero . (Contributed by Thierry Arnoux, 25-Apr-2017.)

Theoremballotlemrc 23105* Range of . (Contributed by Thierry Arnoux, 19-Apr-2017.)

Theoremballotlemirc 23106* Applying does not change first ties. (Contributed by Thierry Arnoux, 19-Apr-2017.)

Theoremballotlemrinv0 23107* Lemma for ballotlemrinv 23108. (Contributed by Thierry Arnoux, 18-Apr-2017.)

Theoremballotlemrinv 23108* is its own inverse : it is an involution. (Contributed by Thierry Arnoux, 10-Apr-2017.)

Theoremballotlem1ri 23109* When the vote on the first tie is for A, the first vote is also for A on the reverse counting. (Contributed by Thierry Arnoux, 18-Apr-2017.)

Theoremballotlem7 23110* is a bijection between two subsets of : one where a vote for A is picked first, and one where a vote for B is picked first (Contributed by Thierry Arnoux, 12-Dec-2016.)

Theoremballotlem8 23111* There are as many countings with ties starting with a ballot for A as there are starting with a ballot for B. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremballotth 23112* Bertrand's ballot problem : the probability that A is ahead throughout the counting. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theorem3o1cs 23113 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Theorem3o2cs 23114 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Theorem3o3cs 23115 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)

18.3.2  Division in the extended real number system

Syntaxcxdiv 23116 Extend class notation to include division of extended reals.
/𝑒

Definitiondf-xdiv 23117* Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.)
/𝑒

Theoremxdivval 23118* Value of division: the (unique) element such that . This is meaningful only when is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
/𝑒

Theoremxrecex 23119* Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.)

Theoremxmulcand 23120 Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)

Theoremxreceu 23121* Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 17-Dec-2016.)

Theoremxdivcld 23122 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
/𝑒

Theoremxdivcl 23123 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
/𝑒

Theoremxdivmul 23124 Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)
/𝑒

Theoremrexdiv 23125 The extended real division operation when both arguments are real. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremxdivrec 23126 Relationship between division and reciprocal. (Contributed by Thierry Arnoux, 5-Jul-2017.)
/𝑒 /𝑒

Theoremxdivid 23127 A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremxdiv0 23128 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremxdiv0rp 23129 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremxgtpnf 23130 An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)

Theoremeliccioo 23131 Membership in a closed interval of extended reals vs. the same open interval (Contributed by Thierry Arnoux, 18-Dec-2016.)

Theoremelxrge02 23132 Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)

Theoremxdivpnfrp 23133 Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremrpxdivcld 23134 Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

Theoremxrpxdivcld 23135 Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
/𝑒

18.3.3  Propositional Calculus - misc additions

Theorembisimpd 23136 Removing of a condition for a biconditionnal connective. (Contributed by Thierry Arnoux, 23-Oct-2016.)

Theorembisimp 23137 Removing of a condition for a biconditionnal connective. (Contributed by Thierry Arnoux, 23-Oct-2016.)

Theorembian1d 23138 Adding a superfluous conjunct in a biconditionnal. (Contributed by Thierry Arnoux, 26-Feb-2017.)

Theoremor3di 23139 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)

Theoremor3dir 23140 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)

18.3.4  Subclass relations - misc additions

Theoremssrd 23141 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Theoremeqrd 23142 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)

Theoremdifneqnul 23143 If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)

Theoremdifeq 23144 Rewriting an equation with set difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)

18.3.5  Restricted Quantification - misc additions

Theoremabeq2f 23145 Equality of a class variable and a class abstraction. In this version, the fact that is a non-free variable in is explicitely stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.)

Theoremeqvincg 23146* A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)

Theoremraleqbid 23147 Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Theoremrexeqbid 23148 Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Theoremralcom4f 23149* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)

Theoremrexcom4f 23150* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)

Theoremrabid2f 23151 An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)

Theoremrabss3d 23152* Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)

Theorem2reuswap2 23153* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.)

Theoremreuxfr3d 23154* Transfer existential uniqueness from a variable to another variable contained in expression . Cf. reuxfr2d 4573 (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Theoremreuxfr4d 23155* Transfer existential uniqueness from a variable to another variable contained in expression . Cf. reuxfrd 4575 (Contributed by Thierry Arnoux, 7-Apr-2017.)

Theoremrexunirn 23156* Restricted existential quantification over the union of the range of a function. Cf. rexrn 5683 and eluni2 3847. (Contributed by Thierry Arnoux, 19-Sep-2017.)

18.3.6  Substitution (without distinct variables) - misc additions

Theoremsbcss12g 23157* Set substitution into the both argument of a subset relation. (Contributed by Thierry Arnoux, 25-Jan-2017.)

Theoremclelsb3f 23158 Substitution applied to an atomic wff (class version of elsb3 2055). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)

18.3.7  Existential Uniqueness - misc additions

Theoremmo5f 23159* Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)

Theoremnmo 23160* Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)

Theoremmoimd 23161* "At most one" is preserved through implication (notice wff reversal). (Contributed by Thierry Arnoux, 25-Feb-2017.)

TheoremrmoxfrdOLD 23162* Transfer "at most one" restricted quantification from a variable to another variable contained in expression . (Contributed by Thierry Arnoux, 7-Apr-2017.)

Theoremrmoxfrd 23163* Transfer "at most one" restricted quantification from a variable to another variable contained in expression . (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Theoremssrmo 23164 "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)

18.3.8  Conditional operator - misc additions

Theoremifbieq12d2 23165 Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.)

Theoremovif 23166 Move a conditional outside of an operation (Contributed by Thierry Arnoux, 25-Jan-2017.)

Theoremelimifd 23167 Elimination of a conditional operator contained in a wff . (Contributed by Thierry Arnoux, 25-Jan-2017.)

Theoremelim2if 23168 Elimination of two conditional operators contained in a wff . (Contributed by Thierry Arnoux, 25-Jan-2017.)

Theoremelim2ifim 23169 Elimination of two conditional operators for an implication. (Contributed by Thierry Arnoux, 25-Jan-2017.)

18.3.9  Indexed union - misc additions

Theoremiuneq12daf 23170 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.)

Theoremiuneq12df 23171 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)

Theoremssiun3 23172* Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)

Theoremssiun2sf 23173 Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)

Theoremiuninc 23174* The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)

Theoremiundifdifd 23175* The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Theoremiundifdif 23176* The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 23175 (Contributed by Thierry Arnoux, 4-Sep-2016.)

Theoremiunrdx 23177* Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)

18.3.10  Miscellaneous

Theoremceqsexv2d 23178* Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)

Theoremr19.41vv 23179* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. Version with two quantifiers (Contributed by Thierry Arnoux, 25-Jan-2017.)

Theoremrabbidva2 23180* Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Theoremabrexdomjm 23181* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremabrexdom2jm 23182* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminin 23183 Intersection with an intersection (Contributed by Thierry Arnoux, 27-Dec-2016.)

Theoremsumpr 23184* A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.)

Theoreminfi1 23185 The interection of two sets is finite if one of them is. (Contributed by Thierry Arnoux, 14-Feb-2017.)

Theoremrabfi 23186* A restricted class built from a finite set is finite. (Contributed by Thierry Arnoux, 14-Feb-2017.)

TheoremrabexgfGS 23187 Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)

Theoremfneq12 23188 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Theoremhashresfn 23189 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Theoremdmhashres 23190 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 12-Jan-2017.)

Theoremcntnevol 23191 Counting and Lebesgue measure are different. (Contributed by Thierry Arnoux, 27-Jan-2017.)

Theoremdisjex 23192* Two ways to say that two classes are disjoint (or equal). (Contributed by Thierry Arnoux, 4-Oct-2016.)

Theoremdisjexc 23193* A variant of disjex 23192, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.)

Theoremrmo3f 23194* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Theoremrmo4fOLD 23195* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.)

Theoremrmo4f 23196* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Theoremr19.29d2r 23197 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version (Contributed by Thierry Arnoux, 30-Jan-2017.)

Theorem19.9d2rf 23198 A deduction version of one direction of 19.9 1795 with two variables (Contributed by Thierry Arnoux, 20-Mar-2017.)

Theorem19.9d2r 23199* A deduction version of one direction of 19.9 1795 with two variables (Contributed by Thierry Arnoux, 30-Jan-2017.)

Theoremprsspwg 23200 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.)

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