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Type | Label | Description |
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Statement | ||
Theorem | tridc 6801* | A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.) |
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Theorem | fimax2gtrilemstep 6802* | Lemma for fimax2gtri 6803. The induction step. (Contributed by Jim Kingdon, 5-Sep-2022.) |
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Theorem | fimax2gtri 6803* | A finite set has a maximum under a trichotomous order. (Contributed by Jim Kingdon, 5-Sep-2022.) |
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Theorem | finexdc 6804* | Decidability of existence, over a finite set and defined by a decidable proposition. (Contributed by Jim Kingdon, 12-Jul-2022.) |
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Theorem | dfrex2fin 6805* | Relationship between universal and existential quantifiers over a finite set. Remark in Section 2.2.1 of [Pierik], p. 8. Although Pierik does not mention the decidability condition explicitly, it does say "only finitely many x to check" which means there must be some way of checking each value of x. (Contributed by Jim Kingdon, 11-Jul-2022.) |
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Theorem | infm 6806* | An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.) |
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Theorem | infn0 6807 | An infinite set is not empty. (Contributed by NM, 23-Oct-2004.) |
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Theorem | inffiexmid 6808* | If any given set is either finite or infinite, excluded middle follows. (Contributed by Jim Kingdon, 15-Jun-2022.) |
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Theorem | en2eqpr 6809 | Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
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Theorem | exmidpw 6810 |
Excluded middle is equivalent to the power set of ![]() |
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Theorem | fientri3 6811 | Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.) |
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Theorem | nnwetri 6812* |
A natural number is well-ordered by ![]() ![]() |
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Theorem | onunsnss 6813 | Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.) |
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Theorem | unfiexmid 6814* | If the union of any two finite sets is finite, excluded middle follows. Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro and Jim Kingdon, 5-Mar-2022.) |
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Theorem | unsnfi 6815 | Adding a singleton to a finite set yields a finite set. (Contributed by Jim Kingdon, 3-Feb-2022.) |
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Theorem | unsnfidcex 6816 |
The ![]() ![]() ![]() |
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Theorem | unsnfidcel 6817 |
The ![]() ![]() ![]() ![]() |
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Theorem | unfidisj 6818 | The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.) |
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Theorem | undifdcss 6819* | Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.) |
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Theorem | undifdc 6820* | Union of complementary parts into whole. This is a case where we can strengthen undifss 3448 from subset to equality. (Contributed by Jim Kingdon, 17-Jun-2022.) |
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Theorem | undiffi 6821 | Union of complementary parts into whole. This is a case where we can strengthen undifss 3448 from subset to equality. (Contributed by Jim Kingdon, 2-Mar-2022.) |
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Theorem | unfiin 6822 | The union of two finite sets is finite if their intersection is. (Contributed by Jim Kingdon, 2-Mar-2022.) |
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Theorem | prfidisj 6823 |
A pair is finite if it consists of two unequal sets. For the case where
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Theorem | tpfidisj 6824 | A triple is finite if it consists of three unequal sets. (Contributed by Jim Kingdon, 1-Oct-2022.) |
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Theorem | fiintim 6825* |
If a class is closed under pairwise intersections, then it is closed
under nonempty finite intersections. The converse would appear to
require an additional condition, such as ![]() ![]() ![]() This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use a pairwise intersection and some texts use a finite intersection, but most topology texts assume excluded middle (in which case the two intersection properties would be equivalent). (Contributed by NM, 22-Sep-2002.) (Revised by Jim Kingdon, 14-Jan-2023.) |
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Theorem | xpfi 6826 | The Cartesian product of two finite sets is finite. Lemma 8.1.16 of [AczelRathjen], p. 74. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.) |
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Theorem | 3xpfi 6827 | The Cartesian product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018.) |
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Theorem | fisseneq 6828 | A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.) |
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Theorem | phpeqd 6829 | Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6767 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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Theorem | ssfirab 6830* | A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.) |
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Theorem | ssfidc 6831* | A subset of a finite set is finite if membership in the subset is decidable. (Contributed by Jim Kingdon, 27-May-2022.) |
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Theorem | snon0 6832 |
An ordinal which is a singleton is ![]() ![]() ![]() |
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Theorem | fnfi 6833 | A version of fnex 5650 for finite sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) |
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Theorem | fundmfi 6834 | The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.) |
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Theorem | fundmfibi 6835 | A function is finite if and only if its domain is finite. (Contributed by AV, 10-Jan-2020.) |
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Theorem | resfnfinfinss 6836 | The restriction of a function to a finite subset of its domain is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.) |
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Theorem | relcnvfi 6837 | If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.) |
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Theorem | funrnfi 6838 | 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 6839 | 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 6840 | A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6841. (Contributed by AV, 10-Jan-2020.) |
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Theorem | f1vrnfibi 6841 | A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 6840. (Contributed by AV, 10-Jan-2020.) |
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Theorem | iunfidisj 6842* |
The finite union of disjoint finite sets is finite. Note that ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | f1finf1o 6843 | 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 6844 | A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) |
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Theorem | en1eqsnbi 6845 | 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 6846* |
A case where the antecedent of snexg 4116 is not needed. The class
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Theorem | preimaf1ofi 6847 | 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 6848* | Lemma for fidcenum 6852. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.) |
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Theorem | fidcenumlemrks 6849* | Lemma for fidcenum 6852. Induction step for fidcenumlemrk 6850. (Contributed by Jim Kingdon, 20-Oct-2022.) |
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Theorem | fidcenumlemrk 6850* | Lemma for fidcenum 6852. (Contributed by Jim Kingdon, 20-Oct-2022.) |
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Theorem | fidcenumlemr 6851* | Lemma for fidcenum 6852. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.) |
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Theorem | fidcenum 6852* |
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 6853* | Lemma for isbth 6863. (Contributed by NM, 22-Mar-1998.) |
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Theorem | sbthlem2 6854* | Lemma for isbth 6863. (Contributed by NM, 22-Mar-1998.) |
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Theorem | sbthlemi3 6855* | Lemma for isbth 6863. (Contributed by NM, 22-Mar-1998.) |
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Theorem | sbthlemi4 6856* | Lemma for isbth 6863. (Contributed by NM, 27-Mar-1998.) |
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Theorem | sbthlemi5 6857* | Lemma for isbth 6863. (Contributed by NM, 22-Mar-1998.) |
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Theorem | sbthlemi6 6858* | Lemma for isbth 6863. (Contributed by NM, 27-Mar-1998.) |
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Theorem | sbthlem7 6859* | Lemma for isbth 6863. (Contributed by NM, 27-Mar-1998.) |
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Theorem | sbthlemi8 6860* | Lemma for isbth 6863. (Contributed by NM, 27-Mar-1998.) |
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Theorem | sbthlemi9 6861* | Lemma for isbth 6863. (Contributed by NM, 28-Mar-1998.) |
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Theorem | sbthlemi10 6862* | Lemma for isbth 6863. (Contributed by NM, 28-Mar-1998.) |
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Theorem | isbth 6863 |
Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This
theorem states that if set ![]() ![]() ![]() ![]() |
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Syntax | cfi 6864 | 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 6865* | 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 6868). (Contributed by FL, 27-Apr-2008.) |
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Theorem | fival 6866* |
The set of all the finite intersections of the elements of ![]() |
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Theorem | elfi 6867* |
Specific properties of an element of ![]() ![]() ![]() ![]() ![]() |
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Theorem | elfi2 6868* | 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 6869 |
Sufficient condition for an element of ![]() ![]() ![]() ![]() ![]() |
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Theorem | ssfii 6870 |
Any element of a set ![]() ![]() |
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Theorem | fi0 6871 | The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.) |
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Theorem | fieq0 6872 | 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 6873 |
Subset relationship for function ![]() |
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Theorem | fiuni 6874 | 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 6875 | 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 6876* | Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.) |
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Syntax | csup 6877 |
Extend class notation to include supremum of class ![]() ![]() ![]() ![]() ![]() |
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Syntax | cinf 6878 |
Extend class notation to include infimum of class ![]() ![]() ![]() ![]() ![]() |
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Definition | df-sup 6879* |
Define the supremum of class ![]() ![]() ![]() |
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Definition | df-inf 6880 |
Define the infimum of class ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | supeq1 6881 | Equality theorem for supremum. (Contributed by NM, 22-May-1999.) |
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Theorem | supeq1d 6882 | Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | supeq1i 6883 | Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | supeq2 6884 | Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | supeq3 6885 | Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.) |
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Theorem | supeq123d 6886 | Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
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Theorem | nfsup 6887 | Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.) |
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Theorem | supmoti 6888* |
Any class ![]() ![]() ![]() |
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Theorem | supeuti 6889* | 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 6890* | Alternate expression for the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.) |
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Theorem | eqsupti 6891* | Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.) |
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Theorem | eqsuptid 6892* | Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 24-Nov-2021.) |
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Theorem | supclti 6893* | A supremum belongs to its base class (closure law). See also supubti 6894 and suplubti 6895. (Contributed by Jim Kingdon, 24-Nov-2021.) |
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Theorem | supubti 6894* |
A supremum is an upper bound. See also supclti 6893 and suplubti 6895.
This proof demonstrates how to expand an iota-based definition (df-iota 5096) using riotacl2 5751. (Contributed by Jim Kingdon, 24-Nov-2021.) |
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Theorem | suplubti 6895* | A supremum is the least upper bound. See also supclti 6893 and supubti 6894. (Contributed by Jim Kingdon, 24-Nov-2021.) |
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Theorem | suplub2ti 6896* | Bidirectional form of suplubti 6895. (Contributed by Jim Kingdon, 17-Jan-2022.) |
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Theorem | supelti 6897* | Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.) |
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Theorem | sup00 6898 | The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
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Theorem | supmaxti 6899* | 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 6900* | The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.) |
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