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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iscard 8801* | Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | iscard2 8802* | Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Theorem | carddom2 8803 | Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 9376, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Theorem | harcard 8804 | The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
har har | ||
Theorem | cardprclem 8805* | Lemma for cardprc 8806. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
Theorem | cardprc 8806 | The class of all cardinal numbers is not a set (i.e. is a proper class). Theorem 19.8 of [Eisenberg] p. 310. In this proof (which does not use AC), we cannot use Cantor's construction canth3 9383 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 8449 to construct (effectively) from , which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.) |
Theorem | carduni 8807* | The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.) |
Theorem | cardiun 8808* | The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.) |
Theorem | cardennn 8809 | If is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.) |
Theorem | cardsucinf 8810 | The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013.) |
Theorem | cardsucnn 8811 | The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 8810. (Contributed by NM, 7-Nov-2008.) |
Theorem | cardom 8812 | The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.) |
Theorem | carden2 8813 | Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 9373, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.) |
Theorem | cardsdom2 8814 | A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Theorem | domtri2 8815 | Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.) |
Theorem | nnsdomel 8816 | Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
Theorem | cardval2 8817* | An alternate version of the value of the cardinal number of a set. Compare cardval 9368. This theorem could be used to give us a simpler definition of in place of df-card 8765. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.) |
Theorem | isinffi 8818* | An infinite set contains subsets equinumerous to every finite set. Extension of isinf 8173 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
Theorem | fidomtri 8819 | Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Theorem | fidomtri2 8820 | Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 7-May-2015.) |
Theorem | harsdom 8821 | The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.) |
har | ||
Theorem | onsdom 8822* | Any well-orderable set is strictly dominated by an ordinal number. (Contributed by Jeff Hankins, 22-Oct-2009.) (Proof shortened by Mario Carneiro, 15-May-2015.) |
Theorem | harval2 8823* | An alternate expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.) |
har | ||
Theorem | cardmin2 8824* | The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.) |
Theorem | pm54.43lem 8825* | In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 8794), so that their means, in our notation, . Here we show that this is equivalent to so that we can use the latter more convenient notation in pm54.43 8826. (Contributed by NM, 4-Nov-2013.) |
Theorem | pm54.43 8826 |
Theorem *54.43 of [WhiteheadRussell]
p. 360. "From this proposition it
will follow, when arithmetical addition has been defined, that
1+1=2."
See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations.
This theorem states that two sets of cardinality 1 are disjoint iff
their union has cardinality 2.
Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 8794), so that their means, in our notation, which is the same as by pm54.43lem 8825. We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.) Theorem pm110.643 8999 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.) |
Theorem | pr2nelem 8827 | Lemma for pr2ne 8828. (Contributed by FL, 17-Aug-2008.) |
Theorem | pr2ne 8828 | If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.) |
Theorem | prdom2 8829 | An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.) |
Theorem | en2eqpr 8830 | Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
Theorem | en2eleq 8831 | Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
Theorem | en2other2 8832 | Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
Theorem | dif1card 8833 | The cardinality of a nonempty finite set is one greater than the cardinality of the set with one element removed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Feb-2013.) |
Theorem | leweon 8834* | Lexicographical order is a well-ordering of . Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 8835, this order is not set-like, as the preimage of is the proper class . (Contributed by Mario Carneiro, 9-Mar-2013.) |
Theorem | r0weon 8835* | A set-like well-ordering of the class of ordinal pairs. Proposition 7.58(1) of [TakeutiZaring] p. 54. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Se | ||
Theorem | infxpenlem 8836* | Lemma for infxpen 8837. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.) |
OrdIso | ||
Theorem | infxpen 8837 | Every infinite ordinal is equinumerous to its Cartesian product. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation is a well-ordering of with the additional property that -initial segments of (where is a limit ordinal) are of cardinality at most . (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Theorem | xpomen 8838 | The Cartesian product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.) |
Theorem | xpct 8839 | The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.) |
Theorem | infxpidm2 8840 | The Cartesian product of an infinite set with itself is idempotent. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 9384. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Theorem | infxpenc 8841* | A canonical version of infxpen 8837, by a completely different approach (although it uses infxpen 8837 via xpomen 8838). Using Cantor's normal form, we can show that respects equinumerosity (oef1o 8595), so that all the steps of can be verified using bijections to do the ordinal commutations. (The assumption on can be satisfied using cnfcom3c 8603.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.) |
finSupp CNF CNF finSupp CNF CNF | ||
Theorem | infxpenc2lem1 8842* | Lemma for infxpenc2 8845. (Contributed by Mario Carneiro, 30-May-2015.) |
Theorem | infxpenc2lem2 8843* | Lemma for infxpenc2 8845. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.) |
finSupp CNF CNF finSupp CNF CNF | ||
Theorem | infxpenc2lem3 8844* | Lemma for infxpenc2 8845. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.) |
Theorem | infxpenc2 8845* | Existence form of infxpenc 8841. A "uniform" or "canonical" version of infxpen 8837, asserting the existence of a single function that simultaneously demonstrates product idempotence of all ordinals below a given bound. (Contributed by Mario Carneiro, 30-May-2015.) |
Theorem | iunmapdisj 8846* | The union is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.) |
Theorem | fseqenlem1 8847* | Lemma for fseqen 8850. (Contributed by Mario Carneiro, 17-May-2015.) |
seq𝜔 | ||
Theorem | fseqenlem2 8848* | Lemma for fseqen 8850. (Contributed by Mario Carneiro, 17-May-2015.) |
seq𝜔 | ||
Theorem | fseqdom 8849* | One half of fseqen 8850. (Contributed by Mario Carneiro, 18-Nov-2014.) |
Theorem | fseqen 8850* | A set that is equinumerous to its Cartesian product is equinumerous to the set of finite sequences on it. (This can be proven more easily using some choice but this proof avoids it.) (Contributed by Mario Carneiro, 18-Nov-2014.) |
Theorem | infpwfidom 8851 | The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption because this theorem also implies that is a set if is.) (Contributed by Mario Carneiro, 17-May-2015.) |
Theorem | dfac8alem 8852* | Lemma for dfac8a 8853. If the power set of a set has a choice function, then the set is numerable. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.) |
recs | ||
Theorem | dfac8a 8853* | Numeration theorem: every set with a choice function on its power set is numerable. With AC, this reduces to the statement that every set is numerable. Similar to Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.) |
Theorem | dfac8b 8854* | The well-ordering theorem: every numerable set is well-orderable. (Contributed by Mario Carneiro, 5-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Theorem | dfac8clem 8855* | Lemma for dfac8c 8856. (Contributed by Mario Carneiro, 10-Jan-2013.) |
Theorem | dfac8c 8856* | If the union of a set is well-orderable, then the set has a choice function. (Contributed by Mario Carneiro, 5-Jan-2013.) |
Theorem | ac10ct 8857* | A proof of the Well ordering theorem weth 9317, an Axiom of Choice equivalent, restricted to sets dominated by some ordinal (in particular finite sets and countable sets), proven in ZF without AC. (Contributed by Mario Carneiro, 5-Jan-2013.) |
Theorem | ween 8858* | A set is numerable iff it can be well-ordered. (Contributed by Mario Carneiro, 5-Jan-2013.) |
Theorem | ac5num 8859* | A version of ac5b 9300 with the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Theorem | ondomen 8860 | If a set is dominated by an ordinal, then it is numerable. (Contributed by Mario Carneiro, 5-Jan-2013.) |
Theorem | numdom 8861 | A set dominated by a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Theorem | ssnum 8862 | A subset of a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Theorem | onssnum 8863 | All subsets of the ordinals are numerable. (Contributed by Mario Carneiro, 12-Feb-2013.) |
Theorem | indcardi 8864* | Indirect strong induction on the cardinality of a finite or numerable set. (Contributed by Stefan O'Rear, 24-Aug-2015.) |
Theorem | acnrcl 8865 | Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC | ||
Theorem | acneq 8866 | Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC AC | ||
Theorem | isacn 8867* | The property of being a choice set of length . (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC | ||
Theorem | acni 8868* | The property of being a choice set of length . (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC | ||
Theorem | acni2 8869* | The property of being a choice set of length . (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC | ||
Theorem | acni3 8870* | The property of being a choice set of length . (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC | ||
Theorem | acnlem 8871* | Construct a mapping satisfying the consequent of isacn 8867. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Theorem | numacn 8872 | A well-orderable set has choice sequences of every length. (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC | ||
Theorem | finacn 8873 | Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC | ||
Theorem | acndom 8874 | A set with long choice sequences also has shorter choice sequences, where "shorter" here means the new index set is dominated by the old index set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC AC | ||
Theorem | acnnum 8875 | A set which has choice sequences on it of length is well-orderable (and hence has choice sequences of every length). (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC | ||
Theorem | acnen 8876 | The class of choice sets of length is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC AC | ||
Theorem | acndom2 8877 | A set smaller than one with choice sequences of length also has choice sequences of length . (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC AC | ||
Theorem | acnen2 8878 | The class of sets with choice sequences of length is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC AC | ||
Theorem | fodomacn 8879 | A version of fodom 9344 that doesn't require the Axiom of Choice ax-ac 9281. If has choice sequences of length , then any surjection from to can be inverted to an injection the other way. (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC | ||
Theorem | fodomnum 8880 | A version of fodom 9344 that doesn't require the Axiom of Choice ax-ac 9281. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fonum 8881 | A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Theorem | numwdom 8882 | A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 27-Aug-2015.) |
* | ||
Theorem | fodomfi2 8883 | Onto functions define dominance when a finite number of choices need to be made. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Theorem | wdomfil 8884 | Weak dominance agrees with normal for finite left sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
* | ||
Theorem | infpwfien 8885 | Any infinite well-orderable set is equinumerous to its set of finite subsets. (Contributed by Mario Carneiro, 18-May-2015.) |
Theorem | inffien 8886 | The set of finite intersections of an infinite well-orderable set is equinumerous to the set itself. (Contributed by Mario Carneiro, 18-May-2015.) |
Theorem | wdomnumr 8887 | Weak dominance agrees with normal for numerable right sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
* | ||
Theorem | alephfnon 8888 | The aleph function is a function on the class of ordinal numbers. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Theorem | aleph0 8889 | The first infinite cardinal number, discovered by Georg Cantor in 1873, has the same size as the set of natural numbers (and under our particular definition is also equal to it). In the literature, the argument of the aleph function is often written as a subscript, and the first aleph is written _0. Exercise 3 of [TakeutiZaring] p. 91. Also Definition 12(i) of [Suppes] p. 228. From Moshé Machover, Set Theory, Logic, and Their Limitations, p. 95: "Aleph...the first letter in the Hebrew alphabet...is also the first letter of the Hebrew word...(einsoph, meaning infinity), which is a cabbalistic appellation of the deity. The notation is due to Cantor, who was deeply interested in mysticism." (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Theorem | alephlim 8890* | Value of the aleph function at a limit ordinal. Definition 12(iii) of [Suppes] p. 91. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Theorem | alephsuc 8891 | Value of the aleph function at a successor ordinal. Definition 12(ii) of [Suppes] p. 91. Here we express the successor aleph in terms of the Hartogs function df-har 8463, which gives the smallest ordinal that strictly dominates its argument (or the supremum of all ordinals that are dominated by the argument). (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
har | ||
Theorem | alephon 8892 | An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Theorem | alephcard 8893 | Every aleph is a cardinal number. Theorem 65 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.) (Revised by Mario Carneiro, 2-Feb-2013.) |
Theorem | alephnbtwn 8894 | No cardinal can be sandwiched between an aleph and its successor aleph. Theorem 67 of [Suppes] p. 229. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 15-May-2015.) |
Theorem | alephnbtwn2 8895 | No set has equinumerosity between an aleph and its successor aleph. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.) |
Theorem | alephordilem1 8896 | Lemma for alephordi 8897. (Contributed by NM, 23-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.) |
Theorem | alephordi 8897 | Strict ordering property of the aleph function. (Contributed by Mario Carneiro, 2-Feb-2013.) |
Theorem | alephord 8898 | Ordering property of the aleph function. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 9-Feb-2013.) |
Theorem | alephord2 8899 | Ordering property of the aleph function. Theorem 8A(a) of [Enderton] p. 213 and its converse. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 9-Feb-2013.) |
Theorem | alephord2i 8900 | Ordering property of the aleph function. Theorem 66 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.) |
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