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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | updjudhcoinlf 9801* | The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022.) |
β’ (π β πΉ:π΄βΆπΆ) & β’ (π β πΊ:π΅βΆπΆ) & β’ π» = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β , (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β β’ (π β (π» β (inl βΎ π΄)) = πΉ) | ||
Theorem | updjudhcoinrg 9802* | The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.) |
β’ (π β πΉ:π΄βΆπΆ) & β’ (π β πΊ:π΅βΆπΆ) & β’ π» = (π₯ β (π΄ β π΅) β¦ if((1st βπ₯) = β , (πΉβ(2nd βπ₯)), (πΊβ(2nd βπ₯)))) β β’ (π β (π» β (inr βΎ π΅)) = πΊ) | ||
Theorem | updjud 9803* | Universal property of the disjoint union. This theorem shows that the disjoint union, together with the left and right injections df-inl 9771 and df-inr 9772, is the coproduct in the category of sets, see Wikipedia "Coproduct", https://en.wikipedia.org/wiki/Coproduct 9772 (25-Aug-2023). This is a special case of Example 1 of coproducts in Section 10.67 of [Adamek] p. 185. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.) |
β’ (π β πΉ:π΄βΆπΆ) & β’ (π β πΊ:π΅βΆπΆ) & β’ (π β π΄ β π) & β’ (π β π΅ β π) β β’ (π β β!β(β:(π΄ β π΅)βΆπΆ β§ (β β (inl βΎ π΄)) = πΉ β§ (β β (inr βΎ π΅)) = πΊ)) | ||
Syntax | ccrd 9804 | Extend class definition to include the cardinal size function. |
class card | ||
Syntax | cale 9805 | Extend class definition to include the aleph function. |
class β΅ | ||
Syntax | ccf 9806 | Extend class definition to include the cofinality function. |
class cf | ||
Syntax | wacn 9807 | The axiom of choice for limited-length sequences. |
class AC π΄ | ||
Definition | df-card 9808* | Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. See cardval 10415 for its value and cardval2 9860 for a simpler version of its value. The principal theorem relating cardinality to equinumerosity is carden 10420. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.) |
β’ card = (π₯ β V β¦ β© {π¦ β On β£ π¦ β π₯}) | ||
Definition | df-aleph 9809 | Define the aleph function. Our definition expresses Definition 12 of [Suppes] p. 229 in a closed form, from which we derive the recursive definition as Theorems aleph0 9935, alephsuc 9937, and alephlim 9936. The aleph function provides a one-to-one, onto mapping from the ordinal numbers to the infinite cardinal numbers. Roughly, any aleph is the smallest infinite cardinal number whose size is strictly greater than any aleph before it. (Contributed by NM, 21-Oct-2003.) |
β’ β΅ = rec(har, Ο) | ||
Definition | df-cf 9810* | Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 10116 for its value and a description. (Contributed by NM, 1-Apr-2004.) |
β’ cf = (π₯ β On β¦ β© {π¦ β£ βπ§(π¦ = (cardβπ§) β§ (π§ β π₯ β§ βπ£ β π₯ βπ’ β π§ π£ β π’))}) | ||
Definition | df-acn 9811* | Define a local and length-limited version of the axiom of choice. The definition of the predicate π β AC π΄ is that for all families of nonempty subsets of π indexed on π΄ (i.e. functions π΄βΆπ« π β {β }), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.) |
β’ AC π΄ = {π₯ β£ (π΄ β V β§ βπ β ((π« π₯ β {β }) βm π΄)βπβπ¦ β π΄ (πβπ¦) β (πβπ¦))} | ||
Theorem | cardf2 9812* | The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 20-Sep-2014.) |
β’ card:{π₯ β£ βπ¦ β On π¦ β π₯}βΆOn | ||
Theorem | cardon 9813 | The cardinal number of a set is an ordinal number. Proposition 10.6(1) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 13-Sep-2013.) |
β’ (cardβπ΄) β On | ||
Theorem | isnum2 9814* | A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.) |
β’ (π΄ β dom card β βπ₯ β On π₯ β π΄) | ||
Theorem | isnumi 9815 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β On β§ π΄ β π΅) β π΅ β dom card) | ||
Theorem | ennum 9816 | Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
β’ (π΄ β π΅ β (π΄ β dom card β π΅ β dom card)) | ||
Theorem | finnum 9817 | Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ (π΄ β Fin β π΄ β dom card) | ||
Theorem | onenon 9818 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
β’ (π΄ β On β π΄ β dom card) | ||
Theorem | tskwe 9819* | A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β π β§ {π₯ β π« π΄ β£ π₯ βΊ π΄} β π΄) β π΄ β dom card) | ||
Theorem | xpnum 9820 | The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ π΅ β dom card) β (π΄ Γ π΅) β dom card) | ||
Theorem | cardval3 9821* | An alternate definition of the value of (cardβπ΄) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.) |
β’ (π΄ β dom card β (cardβπ΄) = β© {π₯ β On β£ π₯ β π΄}) | ||
Theorem | cardid2 9822 | Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) |
β’ (π΄ β dom card β (cardβπ΄) β π΄) | ||
Theorem | isnum3 9823 | A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.) |
β’ (π΄ β dom card β (cardβπ΄) β π΄) | ||
Theorem | oncardval 9824* | The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 10415, this theorem does not require the Axiom of Choice. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
β’ (π΄ β On β (cardβπ΄) = β© {π₯ β On β£ π₯ β π΄}) | ||
Theorem | oncardid 9825 | Any ordinal number is equinumerous to its cardinal number. Unlike cardid 10416, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.) |
β’ (π΄ β On β (cardβπ΄) β π΄) | ||
Theorem | cardonle 9826 | The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) |
β’ (π΄ β On β (cardβπ΄) β π΄) | ||
Theorem | card0 9827 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
β’ (cardββ ) = β | ||
Theorem | cardidm 9828 | The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) |
β’ (cardβ(cardβπ΄)) = (cardβπ΄) | ||
Theorem | oncard 9829* | A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) |
β’ (βπ₯ π΄ = (cardβπ₯) β π΄ = (cardβπ΄)) | ||
Theorem | ficardom 9830 | The cardinal number of a finite set is a finite ordinal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 4-Feb-2013.) |
β’ (π΄ β Fin β (cardβπ΄) β Ο) | ||
Theorem | ficardid 9831 | A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.) |
β’ (π΄ β Fin β (cardβπ΄) β π΄) | ||
Theorem | cardnn 9832 | The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90. (Contributed by Mario Carneiro, 7-Jan-2013.) |
β’ (π΄ β Ο β (cardβπ΄) = π΄) | ||
Theorem | cardnueq0 9833 | The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.) |
β’ (π΄ β dom card β ((cardβπ΄) = β β π΄ = β )) | ||
Theorem | cardne 9834 | No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.) |
β’ (π΄ β (cardβπ΅) β Β¬ π΄ β π΅) | ||
Theorem | carden2a 9835 | If two sets have equal nonzero cardinalities, then they are equinumerous. This assertion and carden2b 9836 are meant to replace carden 10420 in ZF without AC. (Contributed by Mario Carneiro, 9-Jan-2013.) |
β’ (((cardβπ΄) = (cardβπ΅) β§ (cardβπ΄) β β ) β π΄ β π΅) | ||
Theorem | carden2b 9836 | If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 9835 are meant to replace carden 10420 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) |
β’ (π΄ β π΅ β (cardβπ΄) = (cardβπ΅)) | ||
Theorem | card1 9837* | A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.) |
β’ ((cardβπ΄) = 1o β βπ₯ π΄ = {π₯}) | ||
Theorem | cardsn 9838 | A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.) |
β’ (π΄ β π β (cardβ{π΄}) = 1o) | ||
Theorem | carddomi2 9839 | Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10423, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ π΅ β π) β ((cardβπ΄) β (cardβπ΅) β π΄ βΌ π΅)) | ||
Theorem | sdomsdomcardi 9840 | A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.) |
β’ (π΄ βΊ (cardβπ΅) β π΄ βΊ π΅) | ||
Theorem | cardlim 9841 | An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.) |
β’ (Ο β (cardβπ΄) β Lim (cardβπ΄)) | ||
Theorem | cardsdomelir 9842 | A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9843 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.) |
β’ (π΄ β (cardβπ΅) β π΄ βΊ π΅) | ||
Theorem | cardsdomel 9843 | 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.) |
β’ ((π΄ β On β§ π΅ β dom card) β (π΄ βΊ π΅ β π΄ β (cardβπ΅))) | ||
Theorem | iscard 9844* | Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.) |
β’ ((cardβπ΄) = π΄ β (π΄ β On β§ βπ₯ β π΄ π₯ βΊ π΄)) | ||
Theorem | iscard2 9845* | Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.) |
β’ ((cardβπ΄) = π΄ β (π΄ β On β§ βπ₯ β On (π΄ β π₯ β π΄ β π₯))) | ||
Theorem | carddom2 9846 | Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 10423, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΌ π΅)) | ||
Theorem | harcard 9847 | 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.) |
β’ (cardβ(harβπ΄)) = (harβπ΄) | ||
Theorem | cardprclem 9848* | Lemma for cardprc 9849. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
β’ π΄ = {π₯ β£ (cardβπ₯) = π₯} β β’ Β¬ π΄ β V | ||
Theorem | cardprc 9849 | 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 10430 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 9413 to construct (effectively) (β΅βsuc π΄) from (β΅βπ΄), which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.) |
β’ {π₯ β£ (cardβπ₯) = π₯} β V | ||
Theorem | carduni 9850* | The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.) |
β’ (π΄ β π β (βπ₯ β π΄ (cardβπ₯) = π₯ β (cardββͺ π΄) = βͺ π΄)) | ||
Theorem | cardiun 9851* | The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.) |
β’ (π΄ β π β (βπ₯ β π΄ (cardβπ΅) = π΅ β (cardββͺ π₯ β π΄ π΅) = βͺ π₯ β π΄ π΅)) | ||
Theorem | cardennn 9852 | If π΄ is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.) |
β’ ((π΄ β π΅ β§ π΅ β Ο) β (cardβπ΄) = π΅) | ||
Theorem | cardsucinf 9853 | The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013.) |
β’ ((π΄ β On β§ Ο β π΄) β (cardβsuc π΄) = (cardβπ΄)) | ||
Theorem | cardsucnn 9854 | The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 9853. (Contributed by NM, 7-Nov-2008.) |
β’ (π΄ β Ο β (cardβsuc π΄) = suc (cardβπ΄)) | ||
Theorem | cardom 9855 | The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.) |
β’ (cardβΟ) = Ο | ||
Theorem | carden2 9856 | Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 10420, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.) |
β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) | ||
Theorem | cardsdom2 9857 | 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.) |
β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΊ π΅)) | ||
Theorem | domtri2 9858 | Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ π΅ β dom card) β (π΄ βΌ π΅ β Β¬ π΅ βΊ π΄)) | ||
Theorem | nnsdomel 9859 | Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
β’ ((π΄ β Ο β§ π΅ β Ο) β (π΄ β π΅ β π΄ βΊ π΅)) | ||
Theorem | cardval2 9860* | An alternate version of the value of the cardinal number of a set. Compare cardval 10415. This theorem could be used to give a simpler definition of card in place of df-card 9808. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.) |
β’ (π΄ β dom card β (cardβπ΄) = {π₯ β On β£ π₯ βΊ π΄}) | ||
Theorem | isinffi 9861* | An infinite set contains subsets equinumerous to every finite set. Extension of isinf 9137 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
β’ ((Β¬ π΄ β Fin β§ π΅ β Fin) β βπ π:π΅β1-1βπ΄) | ||
Theorem | fidomtri 9862 | 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.) |
β’ ((π΄ β Fin β§ π΅ β π) β (π΄ βΌ π΅ β Β¬ π΅ βΊ π΄)) | ||
Theorem | fidomtri2 9863 | 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.) |
β’ ((π΄ β π β§ π΅ β Fin) β (π΄ βΌ π΅ β Β¬ π΅ βΊ π΄)) | ||
Theorem | harsdom 9864 | The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π΄ β dom card β π΄ βΊ (harβπ΄)) | ||
Theorem | onsdom 9865* | 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.) |
β’ (π΄ β dom card β βπ₯ β On π΄ βΊ π₯) | ||
Theorem | harval2 9866* | An alternate expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π΄ β dom card β (harβπ΄) = β© {π₯ β On β£ π΄ βΊ π₯}) | ||
Theorem | harsucnn 9867 | The next cardinal after a finite ordinal is the successor ordinal. (Contributed by RP, 5-Nov-2023.) |
β’ (π΄ β Ο β (harβπ΄) = suc π΄) | ||
Theorem | cardmin2 9868* | The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.) |
β’ (βπ₯ β On π΄ βΊ π₯ β (cardββ© {π₯ β On β£ π΄ βΊ π₯}) = β© {π₯ β On β£ π΄ βΊ π₯}) | ||
Theorem | pm54.43lem 9869* | 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 9837), so that their π΄ β 1 means, in our notation, π΄ β {π₯ β£ (cardβπ₯) = 1o}. Here we show that this is equivalent to π΄ β 1o so that we can use the latter more convenient notation in pm54.43 9870. (Contributed by NM, 4-Nov-2013.) |
β’ (π΄ β 1o β π΄ β {π₯ β£ (cardβπ₯) = 1o}) | ||
Theorem | pm54.43 9870 |
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 9837), so that their π΄ β 1 means, in our notation, π΄ β {π₯ β£ (cardβπ₯) = 1o} which is the same as π΄ β 1o by pm54.43lem 9869. 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 dju1p1e2 10042 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.) |
β’ ((π΄ β 1o β§ π΅ β 1o) β ((π΄ β© π΅) = β β (π΄ βͺ π΅) β 2o)) | ||
Theorem | enpr2 9871 | An unordered pair with distinct elements is equinumerous to ordinal two. This is a closed-form version of enpr2d 8926. (Contributed by FL, 17-Aug-2008.) Avoid ax-pow 5318, ax-un 7662. (Revised by BTernaryTau, 30-Dec-2024.) |
β’ ((π΄ β πΆ β§ π΅ β π· β§ π΄ β π΅) β {π΄, π΅} β 2o) | ||
Theorem | pr2nelemOLD 9872 | Obsolete version of enpr2 9871 as of 30-Dec-2024. (Contributed by FL, 17-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((π΄ β πΆ β§ π΅ β π· β§ π΄ β π΅) β {π΄, π΅} β 2o) | ||
Theorem | pr2ne 9873 | If an unordered pair has two elements, then they are different. (Contributed by FL, 14-Feb-2010.) Avoid ax-pow 5318, ax-un 7662. (Revised by BTernaryTau, 30-Dec-2024.) |
β’ ((π΄ β πΆ β§ π΅ β π·) β ({π΄, π΅} β 2o β π΄ β π΅)) | ||
Theorem | pr2neOLD 9874 | Obsolete version of pr2ne 9873 as of 30-Dec-2024. (Contributed by FL, 14-Feb-2010.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((π΄ β πΆ β§ π΅ β π·) β ({π΄, π΅} β 2o β π΄ β π΅)) | ||
Theorem | prdom2 9875 | An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.) |
β’ ((π΄ β πΆ β§ π΅ β π·) β {π΄, π΅} βΌ 2o) | ||
Theorem | en2eqpr 9876 | Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
β’ ((πΆ β 2o β§ π΄ β πΆ β§ π΅ β πΆ) β (π΄ β π΅ β πΆ = {π΄, π΅})) | ||
Theorem | en2eleq 9877 | 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.) |
β’ ((π β π β§ π β 2o) β π = {π, βͺ (π β {π})}) | ||
Theorem | en2other2 9878 | Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
β’ ((π β π β§ π β 2o) β βͺ (π β {βͺ (π β {π})}) = π) | ||
Theorem | dif1card 9879 | 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.) |
β’ ((π΄ β Fin β§ π β π΄) β (cardβπ΄) = suc (cardβ(π΄ β {π}))) | ||
Theorem | leweon 9880* | Lexicographical order is a well-ordering of On Γ On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 9881, this order is not set-like, as the preimage of β¨1o, β β© is the proper class ({β } Γ On). (Contributed by Mario Carneiro, 9-Mar-2013.) |
β’ πΏ = {β¨π₯, π¦β© β£ ((π₯ β (On Γ On) β§ π¦ β (On Γ On)) β§ ((1st βπ₯) β (1st βπ¦) β¨ ((1st βπ₯) = (1st βπ¦) β§ (2nd βπ₯) β (2nd βπ¦))))} β β’ πΏ We (On Γ On) | ||
Theorem | r0weon 9881* | 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.) |
β’ πΏ = {β¨π₯, π¦β© β£ ((π₯ β (On Γ On) β§ π¦ β (On Γ On)) β§ ((1st βπ₯) β (1st βπ¦) β¨ ((1st βπ₯) = (1st βπ¦) β§ (2nd βπ₯) β (2nd βπ¦))))} & β’ π = {β¨π§, π€β© β£ ((π§ β (On Γ On) β§ π€ β (On Γ On)) β§ (((1st βπ§) βͺ (2nd βπ§)) β ((1st βπ€) βͺ (2nd βπ€)) β¨ (((1st βπ§) βͺ (2nd βπ§)) = ((1st βπ€) βͺ (2nd βπ€)) β§ π§πΏπ€)))} β β’ (π We (On Γ On) β§ π Se (On Γ On)) | ||
Theorem | infxpenlem 9882* | Lemma for infxpen 9883. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.) |
β’ πΏ = {β¨π₯, π¦β© β£ ((π₯ β (On Γ On) β§ π¦ β (On Γ On)) β§ ((1st βπ₯) β (1st βπ¦) β¨ ((1st βπ₯) = (1st βπ¦) β§ (2nd βπ₯) β (2nd βπ¦))))} & β’ π = {β¨π§, π€β© β£ ((π§ β (On Γ On) β§ π€ β (On Γ On)) β§ (((1st βπ§) βͺ (2nd βπ§)) β ((1st βπ€) βͺ (2nd βπ€)) β¨ (((1st βπ§) βͺ (2nd βπ§)) = ((1st βπ€) βͺ (2nd βπ€)) β§ π§πΏπ€)))} & β’ π = (π β© ((π Γ π) Γ (π Γ π))) & β’ (π β ((π β On β§ βπ β π (Ο β π β (π Γ π) β π)) β§ (Ο β π β§ βπ β π π βΊ π))) & β’ π = ((1st βπ€) βͺ (2nd βπ€)) & β’ π½ = OrdIso(π, (π Γ π)) β β’ ((π΄ β On β§ Ο β π΄) β (π΄ Γ π΄) β π΄) | ||
Theorem | infxpen 9883 | Every infinite ordinal is equinumerous to its Cartesian square. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation π is a well-ordering of (On Γ On) with the additional property that π -initial segments of (π₯ Γ π₯) (where π₯ is a limit ordinal) are of cardinality at most π₯. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.) |
β’ ((π΄ β On β§ Ο β π΄) β (π΄ Γ π΄) β π΄) | ||
Theorem | xpomen 9884 | 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.) |
β’ (Ο Γ Ο) β Ο | ||
Theorem | xpct 9885 | The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.) |
β’ ((π΄ βΌ Ο β§ π΅ βΌ Ο) β (π΄ Γ π΅) βΌ Ο) | ||
Theorem | infxpidm2 9886 | Every infinite well-orderable set is equinumerous to its Cartesian square. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 10431. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ Ο βΌ π΄) β (π΄ Γ π΄) β π΄) | ||
Theorem | infxpenc 9887* | A canonical version of infxpen 9883, by a completely different approach (although it uses infxpen 9883 via xpomen 9884). Using Cantor's normal form, we can show that π΄ βo π΅ respects equinumerosity (oef1o 9567), so that all the steps of (Οβπ) Β· (Οβπ) β Οβ(2π) β (Οβ2)βπ β Οβπ can be verified using bijections to do the ordinal commutations. (The assumption on π can be satisfied using cnfcom3c 9575.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.) |
β’ (π β π΄ β On) & β’ (π β Ο β π΄) & β’ (π β π β (On β 1o)) & β’ (π β πΉ:(Ο βo 2o)β1-1-ontoβΟ) & β’ (π β (πΉββ ) = β ) & β’ (π β π:π΄β1-1-ontoβ(Ο βo π)) & β’ πΎ = (π¦ β {π₯ β ((Ο βo 2o) βm π) β£ π₯ finSupp β } β¦ (πΉ β (π¦ β β‘( I βΎ π)))) & β’ π» = (((Ο CNF π) β πΎ) β β‘((Ο βo 2o) CNF π)) & β’ πΏ = (π¦ β {π₯ β (Ο βm (π Β·o 2o)) β£ π₯ finSupp β } β¦ (( I βΎ Ο) β (π¦ β β‘(π β β‘π)))) & β’ π = (π§ β 2o, π€ β π β¦ ((π Β·o π§) +o π€)) & β’ π = (π§ β 2o, π€ β π β¦ ((2o Β·o π€) +o π§)) & β’ π½ = (((Ο CNF (2o Β·o π)) β πΏ) β β‘(Ο CNF (π Β·o 2o))) & β’ π = (π₯ β (Ο βo π), π¦ β (Ο βo π) β¦ (((Ο βo π) Β·o π₯) +o π¦)) & β’ π = (π₯ β π΄, π¦ β π΄ β¦ β¨(πβπ₯), (πβπ¦)β©) & β’ πΊ = (β‘π β (((π» β π½) β π) β π)) β β’ (π β πΊ:(π΄ Γ π΄)β1-1-ontoβπ΄) | ||
Theorem | infxpenc2lem1 9888* | Lemma for infxpenc2 9891. (Contributed by Mario Carneiro, 30-May-2015.) |
β’ (π β π΄ β On) & β’ (π β βπ β π΄ (Ο β π β βπ€ β (On β 1o)(πβπ):πβ1-1-ontoβ(Ο βo π€))) & β’ π = (β‘(π₯ β (On β 1o) β¦ (Ο βo π₯))βran (πβπ)) β β’ ((π β§ (π β π΄ β§ Ο β π)) β (π β (On β 1o) β§ (πβπ):πβ1-1-ontoβ(Ο βo π))) | ||
Theorem | infxpenc2lem2 9889* | Lemma for infxpenc2 9891. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.) |
β’ (π β π΄ β On) & β’ (π β βπ β π΄ (Ο β π β βπ€ β (On β 1o)(πβπ):πβ1-1-ontoβ(Ο βo π€))) & β’ π = (β‘(π₯ β (On β 1o) β¦ (Ο βo π₯))βran (πβπ)) & β’ (π β πΉ:(Ο βo 2o)β1-1-ontoβΟ) & β’ (π β (πΉββ ) = β ) & β’ πΎ = (π¦ β {π₯ β ((Ο βo 2o) βm π) β£ π₯ finSupp β } β¦ (πΉ β (π¦ β β‘( I βΎ π)))) & β’ π» = (((Ο CNF π) β πΎ) β β‘((Ο βo 2o) CNF π)) & β’ πΏ = (π¦ β {π₯ β (Ο βm (π Β·o 2o)) β£ π₯ finSupp β } β¦ (( I βΎ Ο) β (π¦ β β‘(π β β‘π)))) & β’ π = (π§ β 2o, π€ β π β¦ ((π Β·o π§) +o π€)) & β’ π = (π§ β 2o, π€ β π β¦ ((2o Β·o π€) +o π§)) & β’ π½ = (((Ο CNF (2o Β·o π)) β πΏ) β β‘(Ο CNF (π Β·o 2o))) & β’ π = (π₯ β (Ο βo π), π¦ β (Ο βo π) β¦ (((Ο βo π) Β·o π₯) +o π¦)) & β’ π = (π₯ β π, π¦ β π β¦ β¨((πβπ)βπ₯), ((πβπ)βπ¦)β©) & β’ πΊ = (β‘(πβπ) β (((π» β π½) β π) β π)) β β’ (π β βπβπ β π΄ (Ο β π β (πβπ):(π Γ π)β1-1-ontoβπ)) | ||
Theorem | infxpenc2lem3 9890* | Lemma for infxpenc2 9891. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.) |
β’ (π β π΄ β On) & β’ (π β βπ β π΄ (Ο β π β βπ€ β (On β 1o)(πβπ):πβ1-1-ontoβ(Ο βo π€))) & β’ π = (β‘(π₯ β (On β 1o) β¦ (Ο βo π₯))βran (πβπ)) & β’ (π β πΉ:(Ο βo 2o)β1-1-ontoβΟ) & β’ (π β (πΉββ ) = β ) β β’ (π β βπβπ β π΄ (Ο β π β (πβπ):(π Γ π)β1-1-ontoβπ)) | ||
Theorem | infxpenc2 9891* | Existence form of infxpenc 9887. A "uniform" or "canonical" version of infxpen 9883, asserting the existence of a single function π that simultaneously demonstrates product idempotence of all ordinals below a given bound. (Contributed by Mario Carneiro, 30-May-2015.) |
β’ (π΄ β On β βπβπ β π΄ (Ο β π β (πβπ):(π Γ π)β1-1-ontoβπ)) | ||
Theorem | iunmapdisj 9892* | The union βͺ π β πΆ(π΄ βm π) is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.) |
β’ β*π β πΆ π΅ β (π΄ βm π) | ||
Theorem | fseqenlem1 9893* | Lemma for fseqen 9896. (Contributed by Mario Carneiro, 17-May-2015.) |
β’ (π β π΄ β π) & β’ (π β π΅ β π΄) & β’ (π β πΉ:(π΄ Γ π΄)β1-1-ontoβπ΄) & β’ πΊ = seqΟ((π β V, π β V β¦ (π₯ β (π΄ βm suc π) β¦ ((πβ(π₯ βΎ π))πΉ(π₯βπ)))), {β¨β , π΅β©}) β β’ ((π β§ πΆ β Ο) β (πΊβπΆ):(π΄ βm πΆ)β1-1βπ΄) | ||
Theorem | fseqenlem2 9894* | Lemma for fseqen 9896. (Contributed by Mario Carneiro, 17-May-2015.) |
β’ (π β π΄ β π) & β’ (π β π΅ β π΄) & β’ (π β πΉ:(π΄ Γ π΄)β1-1-ontoβπ΄) & β’ πΊ = seqΟ((π β V, π β V β¦ (π₯ β (π΄ βm suc π) β¦ ((πβ(π₯ βΎ π))πΉ(π₯βπ)))), {β¨β , π΅β©}) & β’ πΎ = (π¦ β βͺ π β Ο (π΄ βm π) β¦ β¨dom π¦, ((πΊβdom π¦)βπ¦)β©) β β’ (π β πΎ:βͺ π β Ο (π΄ βm π)β1-1β(Ο Γ π΄)) | ||
Theorem | fseqdom 9895* | One half of fseqen 9896. (Contributed by Mario Carneiro, 18-Nov-2014.) |
β’ (π΄ β π β (Ο Γ π΄) βΌ βͺ π β Ο (π΄ βm π)) | ||
Theorem | fseqen 9896* | 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.) |
β’ (((π΄ Γ π΄) β π΄ β§ π΄ β β ) β βͺ π β Ο (π΄ βm π) β (Ο Γ π΄)) | ||
Theorem | infpwfidom 9897 | The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (π« π΄ β© Fin) β V because this theorem also implies that π΄ is a set if π« π΄ β© Fin is.) (Contributed by Mario Carneiro, 17-May-2015.) |
β’ ((π« π΄ β© Fin) β V β π΄ βΌ (π« π΄ β© Fin)) | ||
Theorem | dfac8alem 9898* | Lemma for dfac8a 9899. 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.) |
β’ πΉ = recs(πΊ) & β’ πΊ = (π β V β¦ (πβ(π΄ β ran π))) β β’ (π΄ β πΆ β (βπβπ¦ β π« π΄(π¦ β β β (πβπ¦) β π¦) β π΄ β dom card)) | ||
Theorem | dfac8a 9899* | 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.) |
β’ (π΄ β π΅ β (βββπ¦ β π« π΄(π¦ β β β (ββπ¦) β π¦) β π΄ β dom card)) | ||
Theorem | dfac8b 9900* | The well-ordering theorem: every numerable set is well-orderable. (Contributed by Mario Carneiro, 5-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ (π΄ β dom card β βπ₯ π₯ We π΄) |
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