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Theorem List for Metamath Proof Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcarden2b 7801 If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 7800 are meant to replace carden 8373 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Theoremcard1 7802* A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.)

Theoremcardsn 7803 A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)

Theoremcarddomi2 7804 Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 8376, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremsdomsdomcardi 7805 A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremcardlim 7806 An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremcardsdomelir 7807 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 7808 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremcardsdomel 7808 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.)

Theoremiscard 7809* Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremiscard2 7810* Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremcarddom2 7811 Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 8376, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremharcard 7812 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

Theoremcardprclem 7813* Lemma for cardprc 7814. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)

Theoremcardprc 7814 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 8383 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 7460 to construct (effectively) from , which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.)

Theoremcarduni 7815* The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.)

Theoremcardiun 7816* The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)

Theoremcardennn 7817 If is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.)

Theoremcardsucinf 7818 The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013.)

Theoremcardsucnn 7819 The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 7818. (Contributed by NM, 7-Nov-2008.)

Theoremcardom 7820 The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.)

Theoremcarden2 7821 Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 8373, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)

Theoremcardsdom2 7822 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.)

Theoremdomtri2 7823 Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremnnsdomel 7824 Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremcardval2 7825* An alternate version of the value of the cardinal number of a set. Compare cardval 8368. This theorem could be used to give us a simpler definition of in place of df-card 7773. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)

Theoremisinffi 7826* An infinite set contains subsets equinumerous to every finite set. Extension of isinf 7272 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremfidomtri 7827 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.)

Theoremfidomtri2 7828 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.)

Theoremharsdom 7829 The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.)
har

Theoremonsdom 7830* 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.)

Theoremharval2 7831* An alternative expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)
har

Theoremcardmin2 7832* The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)

Theorempm54.43lem 7833* 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 7802), 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 7834. (Contributed by NM, 4-Nov-2013.)

Theorempm54.43 7834 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 7802), so that their means, in our notation, which is the same as by pm54.43lem 7833. 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 8004 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.)

Theorempr2nelem 7835 Lemma for pr2ne 7836. (Contributed by FL, 17-Aug-2008.)

Theorempr2ne 7836 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)

Theoremprdom2 7837 An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.)

Theoremen2eqpr 7838 Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremdif1card 7839 The cardinality of a non-empty 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.)

Theoremleweon 7840* Lexicographical order is a well-ordering of . Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 7841, this order is not set-like, as the preimage of is the proper class . (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoremr0weon 7841* 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

Theoreminfxpenlem 7842* Lemma for infxpen 7843. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
OrdIso

Theoreminfxpen 7843 Every infinite ordinal is equinumerous to its cross 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.)

Theoremxpomen 7844 The cross 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.)

Theoreminfxpidm2 7845 The cross 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 8384. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoreminfxpenc 7846* A canonical version of infxpen 7843, by a completely different approach (although it uses infxpen 7843 via xpomen 7844). Using Cantor's normal form, we can show that respects equinumerosity (oef1o 7602), so that all the steps of can be verified using bijections to do the ordinal commutations. (The assumption on can be satisfied using cnfcom3c 7610.) (Contributed by Mario Carneiro, 30-May-2015.)
CNF CNF                             CNF CNF

Theoreminfxpenc2lem1 7847* Lemma for infxpenc2 7850. (Contributed by Mario Carneiro, 30-May-2015.)

Theoreminfxpenc2lem2 7848* Lemma for infxpenc2 7850. (Contributed by Mario Carneiro, 30-May-2015.)
CNF CNF                             CNF CNF

Theoreminfxpenc2lem3 7849* Lemma for infxpenc2 7850. (Contributed by Mario Carneiro, 30-May-2015.)

Theoreminfxpenc2 7850* Existence form of infxpenc 7846. A "uniform" or "canonical" version of infxpen 7843, 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.)

Theoremiunmapdisj 7851* The union is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.)

Theoremfseqenlem1 7852* Lemma for fseqen 7855. (Contributed by Mario Carneiro, 17-May-2015.)
seq𝜔

Theoremfseqenlem2 7853* Lemma for fseqen 7855. (Contributed by Mario Carneiro, 17-May-2015.)
seq𝜔

Theoremfseqdom 7854* One half of fseqen 7855. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theoremfseqen 7855* A set that is equinumerous to its cross 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.)

Theoreminfpwfidom 7856 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.)

Theoremdfac8alem 7857* Lemma for dfac8a 7858. 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

Theoremdfac8a 7858* 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.)

Theoremdfac8b 7859* The well-ordering theorem: every numerable set is well-orderable. (Contributed by Mario Carneiro, 5-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremdfac8clem 7860* Lemma for dfac8c 7861. (Contributed by Mario Carneiro, 10-Jan-2013.)

Theoremdfac8c 7861* If the union of a set is well-orderable, then the set has a choice function. (Contributed by Mario Carneiro, 5-Jan-2013.)

Theoremac10ct 7862* A proof of the Well ordering theorem weth 8322, 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.)

Theoremween 7863* A set is numerable iff it can be well-ordered. (Contributed by Mario Carneiro, 5-Jan-2013.)

Theoremac5num 7864* A version of ac5b 8305 with the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)

Theoremondomen 7865 If a set is dominated by an ordinal, then it is numerable. (Contributed by Mario Carneiro, 5-Jan-2013.)

Theoremnumdom 7866 A set dominated by a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremssnum 7867 A subset of a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremonssnum 7868 All subsets of the ordinals are numerable. (Contributed by Mario Carneiro, 12-Feb-2013.)

Theoremindcardi 7869* Indirect strong induction on the cardinality of a finite or numerable set. (Contributed by Stefan O'Rear, 24-Aug-2015.)

Theoremacnrcl 7870 Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC

Theoremacneq 7871 Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC AC

Theoremisacn 7872* The property of being a choice set of length . (Contributed by Mario Carneiro, 31-Aug-2015.)
AC

Theoremacni 7873* The property of being a choice set of length . (Contributed by Mario Carneiro, 31-Aug-2015.)
AC

Theoremacni2 7874* The property of being a choice set of length . (Contributed by Mario Carneiro, 31-Aug-2015.)
AC

Theoremacni3 7875* The property of being a choice set of length . (Contributed by Mario Carneiro, 31-Aug-2015.)
AC

Theoremacnlem 7876* Construct a mapping satisfying the consequent of isacn 7872. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremnumacn 7877 A well-orderable set has choice sequences of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC

Theoremfinacn 7878 Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC

Theoremacndom 7879 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

Theoremacnnum 7880 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

Theoremacnen 7881 The class of choice sets of length is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC AC

Theoremacndom2 7882 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

Theoremacnen2 7883 The class of sets with choice sequences of length is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC AC

Theoremfodomacn 7884 A version of fodom 8349 that doesn't require the Axiom of Choice ax-ac 8286. 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

Theoremfodomnum 7885 A version of fodom 8349 that doesn't require the Axiom of Choice ax-ac 8286. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfonum 7886 A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremnumwdom 7887 A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 27-Aug-2015.)
*

Theoremfodomfi2 7888 Onto functions define dominance when a finite number of choices need to be made. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Theoremwdomfil 7889 Weak dominance agrees with normal for finite left sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
*

Theoreminfpwfien 7890 Any infinite well-orderable set is equinumerous to its set of finite subsets. (Contributed by Mario Carneiro, 18-May-2015.)

Theoreminffien 7891 The set of finite intersections of an infinite well-orderable set is equinumerous to the set itself. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremwdomnumr 7892 Weak dominance agrees with normal for numerable right sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
*

Theoremalephfnon 7893 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.)

Theoremaleph0 7894 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.)

Theoremalephlim 7895* 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.)

Theoremalephsuc 7896 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 7473, 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

Theoremalephon 7897 An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremalephcard 7898 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.)

Theoremalephnbtwn 7899 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.)

Theoremalephnbtwn2 7900 No set has equinumerosity between an aleph and its successor aleph. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

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