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Type | Label | Description |
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Statement | ||
Theorem | pfxccatin12lem1 39101 | Lemma 1 for pfxccatin12 39103. Could replace swrdccatin12lem2b 12885. (Contributed by AV, 9-May-2020.) |
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Theorem | pfxccatin12lem2 39102 | Lemma 2 for pfxccatin12 39103. Could replace swrdccatin12lem2 12888. (Contributed by AV, 9-May-2020.) |
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Theorem | pfxccatin12 39103 | The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 12890. (Contributed by AV, 9-May-2020.) |
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Theorem | pfxccat3 39104 | The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. Could replace swrdccat3 12891. (Contributed by AV, 10-May-2020.) |
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Theorem | pfxccatpfx1 39105 | A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.) |
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Theorem | pfxccatpfx2 39106 | A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.) |
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Theorem | pfxccat3a 39107 | A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. Could replace swrdccat3a 12893. (Contributed by AV, 10-May-2020.) |
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Theorem | pfxccatid 39108 | A prefix of a concatenation of length of the first concatenated word is the first word itself. Could replace swrdccatid 12896. (Contributed by AV, 10-May-2020.) |
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Theorem | ccats1pfxeqbi 39109 | A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. Could replace ccats1swrdeqbi 12897. (Contributed by AV, 10-May-2020.) |
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Theorem | pfxccatin12d 39110 | The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12d 12900. (Contributed by AV, 10-May-2020.) |
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Theorem | reuccatpfxs1lem 39111* | Lemma for reuccatpfxs1 39112. Could replace reuccats1lem 12879. (Contributed by AV, 9-May-2020.) |
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Theorem | reuccatpfxs1 39112* | There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. Could replace reuccats1 12880. (Contributed by AV, 10-May-2020.) |
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Theorem | splvalpfx 39113 | Value of the substring replacement operator. (Contributed by AV, 11-May-2020.) |
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Theorem | repswpfx 39114 | A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.) |
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Theorem | cshword2 39115 | Perform a cyclical shift for a word. (Contributed by AV, 11-May-2020.) |
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Theorem | pfxco 39116 | Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.) |
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Additional theorems for classical first-order logic with equality, ZF set theory and theory of real and complex numbers used for proving the theorems for graph theory. | ||
Theorem | elnelall 39117 | A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.) |
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Theorem | clel5 39118* |
Alternate definition of class membership: a class ![]() ![]() ![]() ![]() |
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Theorem | dfss7 39119* |
Alternate definition of subclass relationship: a class ![]() ![]() ![]() ![]() |
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Theorem | sssseq 39120 | If a class is a subclass of another class, the classes are equal iff the other class is a subclass of the first class. (Contributed by AV, 23-Dec-2020.) |
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Theorem | prcssprc 39121 | The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.) |
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Theorem | ralnralall 39122* | A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.) |
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Theorem | falseral0 39123* | A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.) |
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Theorem | ralralimp 39124* | Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.) |
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Theorem | n0rex 39125* | There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.) |
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Theorem | ssn0rex 39126* | There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.) |
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Theorem | elpwdifsn 39127 | A subset of a set is an element of the power set of the difference of the set with a singleton if the subset does not contain the singleton element. (Contributed by AV, 10-Jan-2020.) |
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Theorem | pr1eqbg 39128 | A (proper) pair is equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
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Theorem | pr1nebg 39129 | A (proper) pair is not equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
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Theorem | prelpw 39130 | A pair of two sets belongs to the power class of a class containing those two sets and vice versa. (Contributed by AV, 8-Jan-2020.) |
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Theorem | rexdifpr 39131 | Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.) |
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Theorem | issn 39132* | A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.) |
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Theorem | n0snor2el 39133* | A nonempty set is either a singleton or contains at least two different elements. (Contributed by AV, 20-Sep-2020.) |
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Theorem | opidg 39134 |
The ordered pair ![]() ![]() ![]() ![]() ![]() |
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Theorem | snopeqop 39135 | Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.) |
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Theorem | propeqop 39136 | Equivalence for an ordered pair equal to a pair of ordered pairs. (Contributed by AV, 18-Sep-2020.) |
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Theorem | propssopi 39137 | If a pair of ordered pairs is a subset of an ordered pair, their first components are equal. (Contributed by AV, 20-Sep-2020.) |
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Theorem | ssprss 39138 | A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.) |
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Theorem | ssprsseq 39139 | A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.) |
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Theorem | elpr2elpr 39140* | For an element of an unordered pair which is a subset of a given set, there is another (maybe the same) element of the given set being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.) |
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Theorem | iunopeqop 39141* | Equivalence for an ordered pair equal to an indexed union of singletons of ordered pairs. (Contributed by AV, 20-Sep-2020.) |
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Theorem | otiunsndisjX 39142* | The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.) |
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Theorem | opabn1stprc 39143* | An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wwf. (Contributed by AV, 27-Dec-2020.) |
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Theorem | ralxfrd2 39144* |
Transfer universal quantification from a variable ![]() ![]() ![]() |
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Theorem | rexxfrd2 39145* |
Transfer existence from a variable ![]() ![]() ![]() |
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Theorem | resresdm 39146 | A restriction by an arbitrary set is a restriction by its domain. (Contributed by AV, 16-Nov-2020.) |
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Theorem | resisresindm 39147 |
The restriction of a relation by a set ![]() ![]() |
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Theorem | ssrelrn 39148* | If a relation is a subset of a cartesian product, then for each element of the range of the relation there is an element of the first set of the cartesian product which is related to the element of the range by the relation. (Contributed by AV, 24-Oct-2020.) |
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Theorem | fvifeq 39149 | Equality of function values with conditional arguments, see also fvif 5903. (Contributed by Alexander van der Vekens, 21-May-2018.) |
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Theorem | 2f1fvneq 39150 | If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.) |
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Theorem | f1cofveqaeq 39151 | If the values of a composition of one-to-one functions for two arguments are equal, the arguments themselves must be equal. (Contributed by AV, 3-Feb-2021.) |
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Theorem | f1cofveqaeqALT 39152 | Alternate proof of f1cofveqaeq 39151, 1 essential step shorter, but having more bytes (305 vs. 282). (Contributed by AV, 3-Feb-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | rnfdmpr 39153 |
The range of a one-to-one function ![]() |
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Theorem | imarnf1pr 39154 |
The image of the range of a function ![]() ![]() ![]() ![]() |
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Theorem | funiun 39155* | A function is a union of singletons of ordered pairs indexed by its domain. (Contributed by AV, 18-Sep-2020.) |
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Theorem | funopsn 39156* | If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) |
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Theorem | funop 39157* | An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) |
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Theorem | funop1 39158* | A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) |
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Theorem | f1ssf1 39159 | A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.) |
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Theorem | funsndifnop 39160 | A singleton of an ordered pair is not an ordered pair if the components are different. (Contributed by AV, 23-Sep-2020.) |
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Theorem | funsneqopsn 39161 | A singleton of an ordered pair is an ordered pair of equal singletons if the components are equal. (Contributed by AV, 24-Sep-2020.) |
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Theorem | funsneqop 39162 | A singleton of an ordered pair is an ordered pair if the components are equal. (Contributed by AV, 24-Sep-2020.) |
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Theorem | funsneqopb 39163 | A singleton of an ordered pair is an ordered pair iff the components are equal. (Contributed by AV, 24-Sep-2020.) |
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Theorem | fundmge2nop 39164 | A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 12-Oct-2020.) |
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Theorem | fun2dmnop 39165 | A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 12-Oct-2020.) |
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Theorem | fun2dmnopgexmpl 39166 | A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) |
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Theorem | opabresex0d 39167* | A collection of ordered pairs, the class of all possible second components being a set, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 1-Jan-2021.) |
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Theorem | opabbrfex0d 39168* | A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-Jan-2021.) |
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Theorem | opabresexd 39169* | A collection of ordered pairs, the second component being a function, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) |
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Theorem | opabbrfexd 39170* | A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.) |
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Theorem | opabresex2d 39171* | Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) |
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Theorem | mptmpt2opabbrd 39172* | The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) |
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Theorem | mptmpt2opabovd 39173* | The operation value of a function value of a collection of ordered pairs of related elements (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) |
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Theorem | fpropnf1 39174 | A function, given by an unordered pair of ordered pairs, which is not injective/one-to-one. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) |
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Theorem | riotaeqimp 39175* | If two restricted iota descriptors for an equality are equal, then the terms of the equality are equal. (Contributed by AV, 6-Dec-2020.) |
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Theorem | resfnfinfin 39176 | The restriction of a function by a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.) |
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Theorem | residfi 39177 | A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.) |
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Theorem | cnambpcma 39178 | ((a-b)+c)-a = c-a holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 23-Mar-2018.) |
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Theorem | cnapbmcpd 39179 | ((a+b)-c)+d = ((a+d)+b)-c holds for complex numbers a,b,c,d. (Contributed by Alexander van der Vekens, 23-Mar-2018.) |
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Theorem | 2txmxeqx 39180 | Two times a complex number minus the number itself results in the number itself. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
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Theorem | leaddsuble 39181 | Addition and subtraction on one side of 'less or equal'. (Contributed by Alexander van der Vekens, 18-Mar-2018.) |
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Theorem | 2leaddle2 39182 | If two real numbers are less than a third real number, the sum of the real numbers is less than twice the third real number. (Contributed by Alexander van der Vekens, 21-May-2018.) |
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Theorem | ltnltne 39183 | Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
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Theorem | p1lep2 39184 | A real number increasd by 1 is less than or equal to the number increased by 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
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Theorem | lelttrdi 39185 | If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.) |
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Theorem | ltsubsubaddltsub 39186 | If the result of subtracting two numbers is greater than a number, the result of adding one of these subtracted numbers to the number is less than the result of subtracting the other subtracted number only. (Contributed by Alexander van der Vekens, 9-Jun-2018.) |
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Theorem | zm1nn 39187 | An integer minus 1 is positive under certain circumstances. (Contributed by Alexander van der Vekens, 9-Jun-2018.) |
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Theorem | lesubnn0 39188 | Subtracting a nonnegative integer from a nonnegative integer which is greater than or equal to the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
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Theorem | ltsubnn0 39189 | Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
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Theorem | nn0resubcl 39190 | Closure law for subtraction of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
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Theorem | eluzge0nn0 39191 | If an integer is greater than or equal to a nonnegative integer, then it is a nonnegative integer. (Contributed by Alexander van der Vekens, 27-Aug-2018.) |
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Theorem | ssfz12 39192 | Subset relationship for finite sets of sequential integers. (Contributed by Alexander van der Vekens, 16-Mar-2018.) |
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Theorem | elfz2z 39193 | Membership of an integer in a finite set of sequential integers starting at 0. (Contributed by Alexander van der Vekens, 25-May-2018.) |
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Theorem | 2elfz3nn0 39194 | If there are two elements in a finite set of sequential integers starting at 0, these two elements as well as the upper bound are nonnegative integers. (Contributed by Alexander van der Vekens, 7-Apr-2018.) |
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Theorem | fz0addcom 39195 | The addition of two members of a finite set of sequential integers starting at 0 is commutative. (Contributed by Alexander van der Vekens, 22-May-2018.) (Revised by Alexander van der Vekens, 9-Jun-2018.) |
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Theorem | 2elfz2melfz 39196 | If the sum of two integers of a 0 based finite set of sequential integers is greater than the upper bound, the difference between one of the integers and the difference between the upper bound and the other integer is in the 0 based finite set of sequential integers with the first integer as upper bound. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by Alexander van der Vekens, 31-May-2018.) |
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Theorem | fz0addge0 39197 | The sum of two integers in 0 based finite sets of sequential integers is greater than or equal to zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
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Theorem | elfzlble 39198 | Membership of an integer in a finite set of sequential integers with the integer as upper bound and a lower bound less than or equal to the integer. (Contributed by AV, 21-Oct-2018.) |
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Theorem | elfzelfzlble 39199 | Membership of an element of a finite set of sequential integers in a finite set of sequential integers with the same upper bound and a lower bound less than the upper bound. (Contributed by AV, 21-Oct-2018.) |
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Theorem | subsubelfzo0 39200 | Subtracting a difference from a number which is not less than the difference results in a bounded nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.) |
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