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Type | Label | Description |
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Statement | ||
Theorem | zorn2 10501* | Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set π΄ (with an ordering relation π ) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 10491 through zorn2lem7 10497; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 10497. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
β’ π΄ β V β β’ ((π Po π΄ β§ βπ€((π€ β π΄ β§ π Or π€) β βπ₯ β π΄ βπ§ β π€ (π§π π₯ β¨ π§ = π₯))) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯π π¦) | ||
Theorem | zorn 10502* | Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 10501 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.) |
β’ π΄ β V β β’ (βπ§((π§ β π΄ β§ [β] Or π§) β βͺ π§ β π΄) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) | ||
Theorem | zornn0 10503* | Variant of Zorn's lemma zorn 10502 in which β , the union of the empty chain, is not required to be an element of π΄. (Contributed by Jeff Madsen, 5-Jan-2011.) |
β’ π΄ β V β β’ ((π΄ β β β§ βπ§((π§ β π΄ β§ π§ β β β§ [β] Or π§) β βͺ π§ β π΄)) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) | ||
Theorem | ttukeylem1 10504* | Lemma for ttukey 10513. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β β’ (π β (πΆ β π΄ β (π« πΆ β© Fin) β π΄)) | ||
Theorem | ttukeylem2 10505* | Lemma for ttukey 10513. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β β’ ((π β§ (πΆ β π΄ β§ π· β πΆ)) β π· β π΄) | ||
Theorem | ttukeylem3 10506* | Lemma for ttukey 10513. (Contributed by Mario Carneiro, 11-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) & β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) β β’ ((π β§ πΆ β On) β (πΊβπΆ) = if(πΆ = βͺ πΆ, if(πΆ = β , π΅, βͺ (πΊ β πΆ)), ((πΊββͺ πΆ) βͺ if(((πΊββͺ πΆ) βͺ {(πΉββͺ πΆ)}) β π΄, {(πΉββͺ πΆ)}, β )))) | ||
Theorem | ttukeylem4 10507* | Lemma for ttukey 10513. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) & β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) β β’ (π β (πΊββ ) = π΅) | ||
Theorem | ttukeylem5 10508* | Lemma for ttukey 10513. The πΊ function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) & β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) β β’ ((π β§ (πΆ β On β§ π· β On β§ πΆ β π·)) β (πΊβπΆ) β (πΊβπ·)) | ||
Theorem | ttukeylem6 10509* | Lemma for ttukey 10513. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) & β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) β β’ ((π β§ πΆ β suc (cardβ(βͺ π΄ β π΅))) β (πΊβπΆ) β π΄) | ||
Theorem | ttukeylem7 10510* | Lemma for ttukey 10513. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) & β’ (π β π΅ β π΄) & β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) & β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) β β’ (π β βπ₯ β π΄ (π΅ β π₯ β§ βπ¦ β π΄ Β¬ π₯ β π¦)) | ||
Theorem | ttukey2g 10511* | The TeichmΓΌller-Tukey Lemma ttukey 10513 with a slightly stronger conclusion: we can set up the maximal element of π΄ so that it also contains some given π΅ β π΄ as a subset. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ ((βͺ π΄ β dom card β§ π΅ β π΄ β§ βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β βπ₯ β π΄ (π΅ β π₯ β§ βπ¦ β π΄ Β¬ π₯ β π¦)) | ||
Theorem | ttukeyg 10512* | The TeichmΓΌller-Tukey Lemma ttukey 10513 stated with the "choice" as an antecedent (the hypothesis βͺ π΄ β dom card says that βͺ π΄ is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.) |
β’ ((βͺ π΄ β dom card β§ π΄ β β β§ βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) | ||
Theorem | ttukey 10513* | The TeichmΓΌller-Tukey Lemma, an Axiom of Choice equivalent. If π΄ is a nonempty collection of finite character, then π΄ has a maximal element with respect to inclusion. Here "finite character" means that π₯ β π΄ iff every finite subset of π₯ is in π΄. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ π΄ β V β β’ ((π΄ β β β§ βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) | ||
Theorem | axdclem 10514* | Lemma for axdc 10516. (Contributed by Mario Carneiro, 25-Jan-2013.) |
β’ πΉ = (rec((π¦ β V β¦ (πβ{π§ β£ π¦π₯π§})), π ) βΎ Ο) β β’ ((βπ¦ β π« dom π₯(π¦ β β β (πβπ¦) β π¦) β§ ran π₯ β dom π₯ β§ βπ§(πΉβπΎ)π₯π§) β (πΎ β Ο β (πΉβπΎ)π₯(πΉβsuc πΎ))) | ||
Theorem | axdclem2 10515* | Lemma for axdc 10516. Using the full Axiom of Choice, we can construct a choice function π on π« dom π₯. From this, we can build a sequence πΉ starting at any value π β dom π₯ by repeatedly applying π to the set (πΉβπ₯) (where π₯ is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.) |
β’ πΉ = (rec((π¦ β V β¦ (πβ{π§ β£ π¦π₯π§})), π ) βΎ Ο) β β’ (βπ§ π π₯π§ β (ran π₯ β dom π₯ β βπβπ β Ο (πβπ)π₯(πβsuc π))) | ||
Theorem | axdc 10516* | This theorem derives ax-dc 10441 using ax-ac 10454 and ax-inf 9633. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (Contributed by Mario Carneiro, 25-Jan-2013.) |
β’ ((βπ¦βπ§ π¦π₯π§ β§ ran π₯ β dom π₯) β βπβπ β Ο (πβπ)π₯(πβsuc π)) | ||
Theorem | fodomg 10517 | An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the axiom of choice ac7g 10469. The axiom of choice is not needed for finite sets, see fodomfi 9325. See also fodomnum 10052. (Contributed by NM, 23-Jul-2004.) (Proof shortened by BJ, 20-May-2024.) |
β’ (π΄ β π β (πΉ:π΄βontoβπ΅ β π΅ βΌ π΄)) | ||
Theorem | fodom 10518 | An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.) |
β’ π΄ β V β β’ (πΉ:π΄βontoβπ΅ β π΅ βΌ π΄) | ||
Theorem | dmct 10519 | The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
β’ (π΄ βΌ Ο β dom π΄ βΌ Ο) | ||
Theorem | rnct 10520 | The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
β’ (π΄ βΌ Ο β ran π΄ βΌ Ο) | ||
Theorem | fodomb 10521* | Equivalence of an onto mapping and dominance for a nonempty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.) |
β’ ((π΄ β β β§ βπ π:π΄βontoβπ΅) β (β βΊ π΅ β§ π΅ βΌ π΄)) | ||
Theorem | wdomac 10522 | When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
β’ (π βΌ* π β π βΌ π) | ||
Theorem | brdom3 10523* | Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.) |
β’ π΅ β V β β’ (π΄ βΌ π΅ β βπ(βπ₯β*π¦ π₯ππ¦ β§ βπ₯ β π΄ βπ¦ β π΅ π¦ππ₯)) | ||
Theorem | brdom5 10524* | An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.) |
β’ π΅ β V β β’ (π΄ βΌ π΅ β βπ(βπ₯ β π΅ β*π¦ π₯ππ¦ β§ βπ₯ β π΄ βπ¦ β π΅ π¦ππ₯)) | ||
Theorem | brdom4 10525* | An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
β’ π΅ β V β β’ (π΄ βΌ π΅ β βπ(βπ₯ β π΅ β*π¦ β π΄ π₯ππ¦ β§ βπ₯ β π΄ βπ¦ β π΅ π¦ππ₯)) | ||
Theorem | brdom7disj 10526* | An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
β’ π΄ β V & β’ π΅ β V & β’ (π΄ β© π΅) = β β β’ (π΄ βΌ π΅ β βπ(βπ₯ β π΅ β*π¦ β π΄ {π₯, π¦} β π β§ βπ₯ β π΄ βπ¦ β π΅ {π¦, π₯} β π)) | ||
Theorem | brdom6disj 10527* | An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.) |
β’ π΄ β V & β’ π΅ β V & β’ (π΄ β© π΅) = β β β’ (π΄ βΌ π΅ β βπ(βπ₯ β π΅ β*π¦{π₯, π¦} β π β§ βπ₯ β π΄ βπ¦ β π΅ {π¦, π₯} β π)) | ||
Theorem | fin71ac 10528 | Once we allow AC, the "strongest" definition of finite set becomes equivalent to the "weakest" and the entire hierarchy collapses. (Contributed by Stefan O'Rear, 29-Oct-2014.) |
β’ FinVII = Fin | ||
Theorem | imadomg 10529 | An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.) |
β’ (π΄ β π΅ β (Fun πΉ β (πΉ β π΄) βΌ π΄)) | ||
Theorem | fimact 10530 | The image by a function of a countable set is countable. (Contributed by Thierry Arnoux, 27-Mar-2018.) |
β’ ((π΄ βΌ Ο β§ Fun πΉ) β (πΉ β π΄) βΌ Ο) | ||
Theorem | fnrndomg 10531 | The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.) |
β’ (π΄ β π΅ β (πΉ Fn π΄ β ran πΉ βΌ π΄)) | ||
Theorem | fnct 10532 | If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
β’ ((πΉ Fn π΄ β§ π΄ βΌ Ο) β πΉ βΌ Ο) | ||
Theorem | mptct 10533* | A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
β’ (π΄ βΌ Ο β (π₯ β π΄ β¦ π΅) βΌ Ο) | ||
Theorem | iunfo 10534* | Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.) |
β’ π = βͺ π₯ β π΄ ({π₯} Γ π΅) β β’ (2nd βΎ π):πβontoββͺ π₯ β π΄ π΅ | ||
Theorem | iundom2g 10535* | An upper bound for the cardinality of a disjoint indexed union, with explicit choice principles. π΅ depends on π₯ and should be thought of as π΅(π₯). (Contributed by Mario Carneiro, 1-Sep-2015.) |
β’ π = βͺ π₯ β π΄ ({π₯} Γ π΅) & β’ (π β βͺ π₯ β π΄ (πΆ βm π΅) β AC π΄) & β’ (π β βπ₯ β π΄ π΅ βΌ πΆ) β β’ (π β π βΌ (π΄ Γ πΆ)) | ||
Theorem | iundomg 10536* | An upper bound for the cardinality of an indexed union, with explicit choice principles. π΅ depends on π₯ and should be thought of as π΅(π₯). (Contributed by Mario Carneiro, 1-Sep-2015.) |
β’ π = βͺ π₯ β π΄ ({π₯} Γ π΅) & β’ (π β βͺ π₯ β π΄ (πΆ βm π΅) β AC π΄) & β’ (π β βπ₯ β π΄ π΅ βΌ πΆ) & β’ (π β (π΄ Γ πΆ) β AC βͺ π₯ β π΄ π΅) β β’ (π β βͺ π₯ β π΄ π΅ βΌ (π΄ Γ πΆ)) | ||
Theorem | iundom 10537* | An upper bound for the cardinality of an indexed union. πΆ depends on π₯ and should be thought of as πΆ(π₯). (Contributed by NM, 26-Mar-2006.) |
β’ ((π΄ β π β§ βπ₯ β π΄ πΆ βΌ π΅) β βͺ π₯ β π΄ πΆ βΌ (π΄ Γ π΅)) | ||
Theorem | unidom 10538* | An upper bound for the cardinality of a union. Theorem 10.47 of [TakeutiZaring] p. 98. (Contributed by NM, 25-Mar-2006.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) |
β’ ((π΄ β π β§ βπ₯ β π΄ π₯ βΌ π΅) β βͺ π΄ βΌ (π΄ Γ π΅)) | ||
Theorem | uniimadom 10539* | An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. (Contributed by NM, 25-Mar-2006.) |
β’ π΄ β V & β’ π΅ β V β β’ ((Fun πΉ β§ βπ₯ β π΄ (πΉβπ₯) βΌ π΅) β βͺ (πΉ β π΄) βΌ (π΄ Γ π΅)) | ||
Theorem | uniimadomf 10540* | An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 10539 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) |
β’ β²π₯πΉ & β’ π΄ β V & β’ π΅ β V β β’ ((Fun πΉ β§ βπ₯ β π΄ (πΉβπ₯) βΌ π΅) β βͺ (πΉ β π΄) βΌ (π΄ Γ π΅)) | ||
Theorem | cardval 10541* | The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 9986 for a simpler version of its value. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.) |
β’ π΄ β V β β’ (cardβπ΄) = β© {π₯ β On β£ π₯ β π΄} | ||
Theorem | cardid 10542 | Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.) |
β’ π΄ β V β β’ (cardβπ΄) β π΄ | ||
Theorem | cardidg 10543 | Any set is equinumerous to its cardinal number. Closed theorem form of cardid 10542. (Contributed by David Moews, 1-May-2017.) |
β’ (π΄ β π΅ β (cardβπ΄) β π΄) | ||
Theorem | cardidd 10544 | Any set is equinumerous to its cardinal number. Deduction form of cardid 10542. (Contributed by David Moews, 1-May-2017.) |
β’ (π β π΄ β π΅) β β’ (π β (cardβπ΄) β π΄) | ||
Theorem | cardf 10545 | 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, 13-Sep-2013.) |
β’ card:VβΆOn | ||
Theorem | carden 10546 |
Two sets are equinumerous iff their cardinal numbers are equal. This
important theorem expresses the essential concept behind
"cardinality" or
"size". This theorem appears as Proposition 10.10 of [TakeutiZaring]
p. 85, Theorem 7P of [Enderton] p. 197,
and Theorem 9 of [Suppes] p. 242
(among others). The Axiom of Choice is required for its proof. Related
theorems are hasheni 14308 and the finite-set-only hashen 14307.
This theorem is also known as Hume's Principle. Gottlob Frege's two-volume Grundgesetze der Arithmetik used his Basic Law V to prove this theorem. Unfortunately Basic Law V caused Frege's system to be inconsistent because it was subject to Russell's paradox (see ru 3777). Later scholars have found that Frege primarily used Basic Law V to Hume's Principle. If Basic Law V is replaced by Hume's Principle in Frege's system, much of Frege's work is restored. Grundgesetze der Arithmetik, once Basic Law V is replaced, proves "Frege's theorem" (the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle). See https://plato.stanford.edu/entries/frege-theorem . We take a different approach, using first-order logic and ZFC, to prove the Peano axioms of arithmetic. The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank (see karden 9890). (Contributed by NM, 22-Oct-2003.) |
β’ ((π΄ β πΆ β§ π΅ β π·) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) | ||
Theorem | cardeq0 10547 | Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.) |
β’ (π΄ β π β ((cardβπ΄) = β β π΄ = β )) | ||
Theorem | unsnen 10548 | Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.) |
β’ π΄ β V & β’ π΅ β V β β’ (Β¬ π΅ β π΄ β (π΄ βͺ {π΅}) β suc (cardβπ΄)) | ||
Theorem | carddom 10549 | Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of [Quine] p. 232. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ ((π΄ β π β§ π΅ β π) β ((cardβπ΄) β (cardβπ΅) β π΄ βΌ π΅)) | ||
Theorem | cardsdom 10550 | Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of [Eisenberg] p. 310. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ ((π΄ β π β§ π΅ β π) β ((cardβπ΄) β (cardβπ΅) β π΄ βΊ π΅)) | ||
Theorem | domtri 10551 | Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ ((π΄ β π β§ π΅ β π) β (π΄ βΌ π΅ β Β¬ π΅ βΊ π΄)) | ||
Theorem | entric 10552 | Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of [Suppes] p. 242. (Contributed by NM, 4-Jan-2004.) |
β’ ((π΄ β π β§ π΅ β π) β (π΄ βΊ π΅ β¨ π΄ β π΅ β¨ π΅ βΊ π΄)) | ||
Theorem | entri2 10553 | Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.) |
β’ ((π΄ β π β§ π΅ β π) β (π΄ βΌ π΅ β¨ π΅ βΊ π΄)) | ||
Theorem | entri3 10554 | Trichotomy of dominance. This theorem is equivalent to the Axiom of Choice. Part of Proposition 4.42(d) of [Mendelson] p. 275. (Contributed by NM, 4-Jan-2004.) |
β’ ((π΄ β π β§ π΅ β π) β (π΄ βΌ π΅ β¨ π΅ βΌ π΄)) | ||
Theorem | sdomsdomcard 10555 | A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.) |
β’ (π΄ βΊ π΅ β π΄ βΊ (cardβπ΅)) | ||
Theorem | canth3 10556 | Cantor's theorem in terms of cardinals. This theorem tells us that no matter how large a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.) |
β’ (π΄ β π β (cardβπ΄) β (cardβπ« π΄)) | ||
Theorem | infxpidm 10557 | Every infinite class is equinumerous to its Cartesian square. This theorem, which is equivalent to the axiom of choice over ZF, provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This is a corollary of infxpen 10009 (used via infxpidm2 10012). (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.) |
β’ (Ο βΌ π΄ β (π΄ Γ π΄) β π΄) | ||
Theorem | ondomon 10558* | The class of ordinals dominated by a given set is an ordinal. Theorem 56 of [Suppes] p. 227. This theorem can be proved without the axiom of choice, see hartogs 9539. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.) Use hartogs 9539 instead. (New usage is discouraged.) |
β’ (π΄ β π β {π₯ β On β£ π₯ βΌ π΄} β On) | ||
Theorem | cardmin 10559* | The smallest ordinal that strictly dominates a set is a cardinal. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 20-Sep-2014.) |
β’ (π΄ β π β (cardββ© {π₯ β On β£ π΄ βΊ π₯}) = β© {π₯ β On β£ π΄ βΊ π₯}) | ||
Theorem | ficard 10560 | A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.) |
β’ (π΄ β π β (π΄ β Fin β (cardβπ΄) β Ο)) | ||
Theorem | infinf 10561 | Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.) |
β’ (π΄ β π΅ β (Β¬ π΄ β Fin β Ο βΌ π΄)) | ||
Theorem | unirnfdomd 10562 | The union of the range of a function from an infinite set into the class of finite sets is dominated by its domain. Deduction form. (Contributed by David Moews, 1-May-2017.) |
β’ (π β πΉ:πβΆFin) & β’ (π β Β¬ π β Fin) & β’ (π β π β π) β β’ (π β βͺ ran πΉ βΌ π) | ||
Theorem | konigthlem 10563* | Lemma for konigth 10564. (Contributed by Mario Carneiro, 22-Feb-2013.) |
β’ π΄ β V & β’ π = βͺ π β π΄ (πβπ) & β’ π = Xπ β π΄ (πβπ) & β’ π· = (π β π΄ β¦ (π β (πβπ) β¦ ((πβπ)βπ))) & β’ πΈ = (π β π΄ β¦ (πβπ)) β β’ (βπ β π΄ (πβπ) βΊ (πβπ) β π βΊ π) | ||
Theorem | konigth 10564* | Konig's Theorem. If π(π) βΊ π(π) for all π β π΄, then Ξ£π β π΄π(π) βΊ βπ β π΄π(π), where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with unions rather than disjoint unions for convenience, but the version with disjoint unions is clearly a special case of this version. The Axiom of Choice is needed for this proof, but it contains AC as a simple corollary (letting π(π) = β , this theorem says that an infinite cartesian product of nonempty sets is nonempty), so this is an AC equivalent. Theorem 11.26 of [TakeutiZaring] p. 107. (Contributed by Mario Carneiro, 22-Feb-2013.) |
β’ π΄ β V & β’ π = βͺ π β π΄ (πβπ) & β’ π = Xπ β π΄ (πβπ) β β’ (βπ β π΄ (πβπ) βΊ (πβπ) β π βΊ π) | ||
Theorem | alephsucpw 10565 | The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 10671 or gchaleph2 10667.) (Contributed by NM, 27-Aug-2005.) |
β’ (β΅βsuc π΄) βΌ π« (β΅βπ΄) | ||
Theorem | aleph1 10566 | The set exponentiation of 2 to the aleph-zero has cardinality of at least aleph-one. (If we were to assume the Continuum Hypothesis, their cardinalities would be the same.) (Contributed by NM, 7-Jul-2004.) |
β’ (β΅β1o) βΌ (2o βm (β΅ββ )) | ||
Theorem | alephval2 10567* | An alternate way to express the value of the aleph function for nonzero arguments. Theorem 64 of [Suppes] p. 229. (Contributed by NM, 15-Nov-2003.) |
β’ ((π΄ β On β§ β β π΄) β (β΅βπ΄) = β© {π₯ β On β£ βπ¦ β π΄ (β΅βπ¦) βΊ π₯}) | ||
Theorem | dominfac 10568 | A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 10454. See dominf 10440 for a version proved from ax-cc 10430. (Contributed by NM, 25-Mar-2007.) |
β’ π΄ β V β β’ ((π΄ β β β§ π΄ β βͺ π΄) β Ο βΌ π΄) | ||
Theorem | iunctb 10569* | The countable union of countable sets is countable (indexed union version of unictb 10570). (Contributed by Mario Carneiro, 18-Jan-2014.) |
β’ ((π΄ βΌ Ο β§ βπ₯ β π΄ π΅ βΌ Ο) β βͺ π₯ β π΄ π΅ βΌ Ο) | ||
Theorem | unictb 10570* | The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 10569 for indexed union version. (Contributed by NM, 26-Mar-2006.) |
β’ ((π΄ βΌ Ο β§ βπ₯ β π΄ π₯ βΌ Ο) β βͺ π΄ βΌ Ο) | ||
Theorem | infmap 10571* | An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. (Contributed by NM, 1-Oct-2004.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) |
β’ ((Ο βΌ π΄ β§ π΅ βΌ π΄) β (π΄ βm π΅) β {π₯ β£ (π₯ β π΄ β§ π₯ β π΅)}) | ||
Theorem | alephadd 10572 | The sum of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ ((β΅βπ΄) β (β΅βπ΅)) β ((β΅βπ΄) βͺ (β΅βπ΅)) | ||
Theorem | alephmul 10573 | The product of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ ((π΄ β On β§ π΅ β On) β ((β΅βπ΄) Γ (β΅βπ΅)) β ((β΅βπ΄) βͺ (β΅βπ΅))) | ||
Theorem | alephexp1 10574 | An exponentiation law for alephs. Lemma 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ (((π΄ β On β§ π΅ β On) β§ π΄ β π΅) β ((β΅βπ΄) βm (β΅βπ΅)) β (2o βm (β΅βπ΅))) | ||
Theorem | alephsuc3 10575* | An alternate representation of a successor aleph. Compare alephsuc 10063 and alephsuc2 10075. Equality can be obtained by taking the card of the right-hand side then using alephcard 10065 and carden 10546. (Contributed by NM, 23-Oct-2004.) |
β’ (π΄ β On β (β΅βsuc π΄) β {π₯ β On β£ π₯ β (β΅βπ΄)}) | ||
Theorem | alephexp2 10576* | An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 10574 (which works if the base is less than or equal to the exponent) and infmap 10571 (which works if the exponent is less than or equal to the base). They can be equated only when the base is equal to the exponent, and this is the result. (Contributed by NM, 23-Oct-2004.) |
β’ (π΄ β On β (2o βm (β΅βπ΄)) β {π₯ β£ (π₯ β (β΅βπ΄) β§ π₯ β (β΅βπ΄))}) | ||
Theorem | alephreg 10577 | A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.) |
β’ (cfβ(β΅βsuc π΄)) = (β΅βsuc π΄) | ||
Theorem | pwcfsdom 10578* | A corollary of Konig's Theorem konigth 10564. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.) |
β’ π» = (π¦ β (cfβ(β΅βπ΄)) β¦ (harβ(πβπ¦))) β β’ (β΅βπ΄) βΊ ((β΅βπ΄) βm (cfβ(β΅βπ΄))) | ||
Theorem | cfpwsdom 10579 | A corollary of Konig's Theorem konigth 10564. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.) |
β’ π΅ β V β β’ (2o βΌ π΅ β (β΅βπ΄) βΊ (cfβ(cardβ(π΅ βm (β΅βπ΄))))) | ||
Theorem | alephom 10580 | From canth2 9130, we know that (β΅β0) < (2βΟ), but we cannot prove that (2βΟ) = (β΅β1) (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement (β΅βπ΄) < (2βΟ) is consistent for any ordinal π΄). However, we can prove that (2βΟ) is not equal to (β΅βΟ), nor (β΅β(β΅βΟ)), on cofinality grounds, because by Konig's Theorem konigth 10564 (in the form of cfpwsdom 10579), (2βΟ) has uncountable cofinality, which eliminates limit alephs like (β΅βΟ). (The first limit aleph that is not eliminated is (β΅β(β΅β1)), which has cofinality (β΅β1).) (Contributed by Mario Carneiro, 21-Mar-2013.) |
β’ (cardβ(2o βm Ο)) β (β΅βΟ) | ||
Theorem | smobeth 10581 | The beth function is strictly monotone. This function is not strictly the beth function, but rather bethA is the same as (cardβ(π 1β(Ο +o π΄))), since conventionally we start counting at the first infinite level, and ignore the finite levels. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 2-Jun-2015.) |
β’ Smo (card β π 1) | ||
Theorem | nd1 10582 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.) |
β’ (βπ₯ π₯ = π¦ β Β¬ βπ₯ π¦ β π§) | ||
Theorem | nd2 10583 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.) |
β’ (βπ₯ π₯ = π¦ β Β¬ βπ₯ π§ β π¦) | ||
Theorem | nd3 10584 | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.) |
β’ (βπ₯ π₯ = π¦ β Β¬ βπ§ π₯ β π¦) | ||
Theorem | nd4 10585 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
β’ (βπ₯ π₯ = π¦ β Β¬ βπ§ π¦ β π₯) | ||
Theorem | axextnd 10586 | A version of the Axiom of Extensionality with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 14-Aug-2003.) (New usage is discouraged.) |
β’ βπ₯((π₯ β π¦ β π₯ β π§) β π¦ = π§) | ||
Theorem | axrepndlem1 10587* | Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
β’ (Β¬ βπ¦ π¦ = π§ β βπ₯(βπ¦βπ§(π β π§ = π¦) β βπ§(π§ β π₯ β βπ₯(π₯ β π¦ β§ βπ¦π)))) | ||
Theorem | axrepndlem2 10588 | Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (New usage is discouraged.) |
β’ (((Β¬ βπ₯ π₯ = π¦ β§ Β¬ βπ₯ π₯ = π§) β§ Β¬ βπ¦ π¦ = π§) β βπ₯(βπ¦βπ§(π β π§ = π¦) β βπ§(π§ β π₯ β βπ₯(π₯ β π¦ β§ βπ¦π)))) | ||
Theorem | axrepnd 10589 | A version of the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
β’ βπ₯(βπ¦βπ§(π β π§ = π¦) β βπ§(βπ¦ π§ β π₯ β βπ₯(βπ§ π₯ β π¦ β§ βπ¦π))) | ||
Theorem | axunndlem1 10590* | Lemma for the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
β’ βπ₯βπ¦(βπ₯(π¦ β π₯ β§ π₯ β π§) β π¦ β π₯) | ||
Theorem | axunnd 10591 | A version of the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
β’ βπ₯βπ¦(βπ₯(π¦ β π₯ β§ π₯ β π§) β π¦ β π₯) | ||
Theorem | axpowndlem1 10592 | Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) |
β’ (βπ₯ π₯ = π¦ β (Β¬ π₯ = π¦ β βπ₯βπ¦(βπ₯(βπ§ π₯ β π¦ β βπ¦ π₯ β π§) β π¦ β π₯))) | ||
Theorem | axpowndlem2 10593* | Lemma for the Axiom of Power Sets with no distinct variable conditions. Revised to remove a redundant antecedent from the consequence. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (Revised and shortened by Wolf Lammen, 9-Jun-2019.) (New usage is discouraged.) |
β’ (Β¬ βπ₯ π₯ = π¦ β (Β¬ βπ₯ π₯ = π§ β βπ₯βπ¦(βπ₯(βπ§ π₯ β π¦ β βπ¦ π₯ β π§) β π¦ β π₯))) | ||
Theorem | axpowndlem3 10594* | Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.) (Proof shortened by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.) |
β’ (Β¬ π₯ = π¦ β βπ₯βπ¦(βπ₯(βπ§ π₯ β π¦ β βπ¦ π₯ β π§) β π¦ β π₯)) | ||
Theorem | axpowndlem4 10595 | Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.) (New usage is discouraged.) |
β’ (Β¬ βπ¦ π¦ = π₯ β (Β¬ βπ¦ π¦ = π§ β (Β¬ π₯ = π¦ β βπ₯βπ¦(βπ₯(βπ§ π₯ β π¦ β βπ¦ π₯ β π§) β π¦ β π₯)))) | ||
Theorem | axpownd 10596 | A version of the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 4-Jan-2002.) (New usage is discouraged.) |
β’ (Β¬ π₯ = π¦ β βπ₯βπ¦(βπ₯(βπ§ π₯ β π¦ β βπ¦ π₯ β π§) β π¦ β π₯)) | ||
Theorem | axregndlem1 10597 | Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.) |
β’ (βπ₯ π₯ = π§ β (π₯ β π¦ β βπ₯(π₯ β π¦ β§ βπ§(π§ β π₯ β Β¬ π§ β π¦)))) | ||
Theorem | axregndlem2 10598* | Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.) (New usage is discouraged.) |
β’ (π₯ β π¦ β βπ₯(π₯ β π¦ β§ βπ§(π§ β π₯ β Β¬ π§ β π¦))) | ||
Theorem | axregnd 10599 | A version of the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) (New usage is discouraged.) |
β’ (π₯ β π¦ β βπ₯(π₯ β π¦ β§ βπ§(π§ β π₯ β Β¬ π§ β π¦))) | ||
Theorem | axinfndlem1 10600* | Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.) |
β’ (βπ₯ π¦ β π§ β βπ₯(π¦ β π₯ β§ βπ¦(π¦ β π₯ β βπ§(π¦ β π§ β§ π§ β π₯)))) |
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