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Type | Label | Description |
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Statement | ||
Theorem | funrnfi 7001 | The range of a finite relation is finite if its converse is a function. (Contributed by Jim Kingdon, 5-Feb-2022.) |
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Theorem | f1ofi 7002 | If a 1-1 and onto function has a finite domain, its range is finite. (Contributed by Jim Kingdon, 21-Feb-2022.) |
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Theorem | f1dmvrnfibi 7003 | A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 7004. (Contributed by AV, 10-Jan-2020.) |
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Theorem | f1vrnfibi 7004 | A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 7003. (Contributed by AV, 10-Jan-2020.) |
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Theorem | iunfidisj 7005* |
The finite union of disjoint finite sets is finite. Note that ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | f1finf1o 7006 | Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) |
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Theorem | en1eqsn 7007 | A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) |
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Theorem | en1eqsnbi 7008 | A set containing an element has exactly one element iff it is a singleton. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
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Theorem | snexxph 7009* |
A case where the antecedent of snexg 4213 is not needed. The class
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Theorem | preimaf1ofi 7010 | The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.) |
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Theorem | fidcenumlemim 7011* | Lemma for fidcenum 7015. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.) |
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Theorem | fidcenumlemrks 7012* | Lemma for fidcenum 7015. Induction step for fidcenumlemrk 7013. (Contributed by Jim Kingdon, 20-Oct-2022.) |
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Theorem | fidcenumlemrk 7013* | Lemma for fidcenum 7015. (Contributed by Jim Kingdon, 20-Oct-2022.) |
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Theorem | fidcenumlemr 7014* | Lemma for fidcenum 7015. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.) |
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Theorem | fidcenum 7015* |
A set is finite if and only if it has decidable equality and is finitely
enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The
definition of "finitely enumerable" as
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Theorem | sbthlem1 7016* | Lemma for isbth 7026. (Contributed by NM, 22-Mar-1998.) |
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Theorem | sbthlem2 7017* | Lemma for isbth 7026. (Contributed by NM, 22-Mar-1998.) |
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Theorem | sbthlemi3 7018* | Lemma for isbth 7026. (Contributed by NM, 22-Mar-1998.) |
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Theorem | sbthlemi4 7019* | Lemma for isbth 7026. (Contributed by NM, 27-Mar-1998.) |
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Theorem | sbthlemi5 7020* | Lemma for isbth 7026. (Contributed by NM, 22-Mar-1998.) |
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Theorem | sbthlemi6 7021* | Lemma for isbth 7026. (Contributed by NM, 27-Mar-1998.) |
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Theorem | sbthlem7 7022* | Lemma for isbth 7026. (Contributed by NM, 27-Mar-1998.) |
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Theorem | sbthlemi8 7023* | Lemma for isbth 7026. (Contributed by NM, 27-Mar-1998.) |
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Theorem | sbthlemi9 7024* | Lemma for isbth 7026. (Contributed by NM, 28-Mar-1998.) |
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Theorem | sbthlemi10 7025* | Lemma for isbth 7026. (Contributed by NM, 28-Mar-1998.) |
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Theorem | isbth 7026 |
Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This
theorem states that if set ![]() ![]() ![]() ![]() |
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Syntax | cfi 7027 | Extend class notation with the function whose value is the class of finite intersections of the elements of a given set. |
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Definition | df-fi 7028* | Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 7031). (Contributed by FL, 27-Apr-2008.) |
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Theorem | fival 7029* |
The set of all the finite intersections of the elements of ![]() |
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Theorem | elfi 7030* |
Specific properties of an element of ![]() ![]() ![]() ![]() ![]() |
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Theorem | elfi2 7031* | The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.) |
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Theorem | elfir 7032 |
Sufficient condition for an element of ![]() ![]() ![]() ![]() ![]() |
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Theorem | ssfii 7033 |
Any element of a set ![]() ![]() |
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Theorem | fi0 7034 | The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.) |
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Theorem | fieq0 7035 | A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.) |
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Theorem | fiss 7036 |
Subset relationship for function ![]() |
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Theorem | fiuni 7037 | The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
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Theorem | fipwssg 7038 | If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
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Theorem | fifo 7039* | Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.) |
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Theorem | dcfi 7040* | Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.) |
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Syntax | csup 7041 |
Extend class notation to include supremum of class ![]() ![]() ![]() ![]() ![]() |
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Syntax | cinf 7042 |
Extend class notation to include infimum of class ![]() ![]() ![]() ![]() ![]() |
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Definition | df-sup 7043* |
Define the supremum of class ![]() ![]() ![]() |
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Definition | df-inf 7044 |
Define the infimum of class ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | supeq1 7045 | Equality theorem for supremum. (Contributed by NM, 22-May-1999.) |
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Theorem | supeq1d 7046 | Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | supeq1i 7047 | Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | supeq2 7048 | Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | supeq3 7049 | Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.) |
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Theorem | supeq123d 7050 | Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
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Theorem | nfsup 7051 | Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.) |
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Theorem | supmoti 7052* |
Any class ![]() ![]() ![]() |
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Theorem | supeuti 7053* | A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by Jim Kingdon, 23-Nov-2021.) |
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Theorem | supval2ti 7054* | Alternate expression for the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.) |
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Theorem | eqsupti 7055* | Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.) |
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Theorem | eqsuptid 7056* | Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 24-Nov-2021.) |
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Theorem | supclti 7057* | A supremum belongs to its base class (closure law). See also supubti 7058 and suplubti 7059. (Contributed by Jim Kingdon, 24-Nov-2021.) |
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Theorem | supubti 7058* |
A supremum is an upper bound. See also supclti 7057 and suplubti 7059.
This proof demonstrates how to expand an iota-based definition (df-iota 5215) using riotacl2 5887. (Contributed by Jim Kingdon, 24-Nov-2021.) |
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Theorem | suplubti 7059* | A supremum is the least upper bound. See also supclti 7057 and supubti 7058. (Contributed by Jim Kingdon, 24-Nov-2021.) |
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Theorem | suplub2ti 7060* | Bidirectional form of suplubti 7059. (Contributed by Jim Kingdon, 17-Jan-2022.) |
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Theorem | supelti 7061* | Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.) |
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Theorem | sup00 7062 | The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
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Theorem | supmaxti 7063* | The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.) |
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Theorem | supsnti 7064* | The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.) |
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Theorem | isotilem 7065* | Lemma for isoti 7066. (Contributed by Jim Kingdon, 26-Nov-2021.) |
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Theorem | isoti 7066* | An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.) |
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Theorem | supisolem 7067* | Lemma for supisoti 7069. (Contributed by Mario Carneiro, 24-Dec-2016.) |
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Theorem | supisoex 7068* | Lemma for supisoti 7069. (Contributed by Mario Carneiro, 24-Dec-2016.) |
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Theorem | supisoti 7069* | Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.) |
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Theorem | infeq1 7070 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | infeq1d 7071 | Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | infeq1i 7072 | Equality inference for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | infeq2 7073 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | infeq3 7074 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | infeq123d 7075 | Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | nfinf 7076 | Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | cnvinfex 7077* | Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.) |
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Theorem | cnvti 7078* | If a relation satisfies a condition corresponding to tightness of an apartness generated by an order, so does its converse. (Contributed by Jim Kingdon, 17-Dec-2021.) |
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Theorem | eqinfti 7079* | Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.) |
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Theorem | eqinftid 7080* | Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.) |
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Theorem | infvalti 7081* | Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.) |
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Theorem | infclti 7082* | An infimum belongs to its base class (closure law). See also inflbti 7083 and infglbti 7084. (Contributed by Jim Kingdon, 17-Dec-2021.) |
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Theorem | inflbti 7083* | An infimum is a lower bound. See also infclti 7082 and infglbti 7084. (Contributed by Jim Kingdon, 18-Dec-2021.) |
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Theorem | infglbti 7084* | An infimum is the greatest lower bound. See also infclti 7082 and inflbti 7083. (Contributed by Jim Kingdon, 18-Dec-2021.) |
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Theorem | infnlbti 7085* | A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.) |
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Theorem | infminti 7086* | The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.) |
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Theorem | infmoti 7087* |
Any class ![]() ![]() ![]() |
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Theorem | infeuti 7088* | An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.) |
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Theorem | infsnti 7089* | The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.) |
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Theorem | inf00 7090 | The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
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Theorem | infisoti 7091* | Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.) |
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Theorem | supex2g 7092 | Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | infex2g 7093 | Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.) |
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Theorem | ordiso2 7094 | Generalize ordiso 7095 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.) |
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Theorem | ordiso 7095* | Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.) |
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Syntax | cdju 7096 | Extend class notation to include disjoint union of two classes. |
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Definition | df-dju 7097 |
Disjoint union of two classes. This is a way of creating a class which
contains elements corresponding to each element of ![]() ![]() ![]() ![]() |
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Theorem | djueq12 7098 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
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Theorem | djueq1 7099 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
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Theorem | djueq2 7100 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
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