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Type | Label | Description |
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Statement | ||
Theorem | r1elss 9801 | The range of the π 1 function is transitive. Lemma 2.10 of [Kunen] p. 97. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
β’ π΄ β V β β’ (π΄ β βͺ (π 1 β On) β π΄ β βͺ (π 1 β On)) | ||
Theorem | pwwf 9802 | A power set is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
β’ (π΄ β βͺ (π 1 β On) β π« π΄ β βͺ (π 1 β On)) | ||
Theorem | sswf 9803 | A subset of a well-founded set is well-founded. (Contributed by Mario Carneiro, 17-Nov-2014.) |
β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β π΄) β π΅ β βͺ (π 1 β On)) | ||
Theorem | snwf 9804 | A singleton is well-founded if its element is. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
β’ (π΄ β βͺ (π 1 β On) β {π΄} β βͺ (π 1 β On)) | ||
Theorem | unwf 9805 | A binary union is well-founded iff its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β βͺ (π 1 β On)) β (π΄ βͺ π΅) β βͺ (π 1 β On)) | ||
Theorem | prwf 9806 | An unordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β βͺ (π 1 β On)) β {π΄, π΅} β βͺ (π 1 β On)) | ||
Theorem | opwf 9807 | An ordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) |
β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β βͺ (π 1 β On)) β β¨π΄, π΅β© β βͺ (π 1 β On)) | ||
Theorem | unir1 9808 | The cumulative hierarchy of sets covers the universe. Proposition 4.45 (b) to (a) of [Mendelson] p. 281. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 8-Jun-2013.) |
β’ βͺ (π 1 β On) = V | ||
Theorem | jech9.3 9809 | Every set belongs to some value of the cumulative hierarchy of sets function π 1, i.e. the indexed union of all values of π 1 is the universe. Lemma 9.3 of [Jech] p. 71. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 8-Jun-2013.) |
β’ βͺ π₯ β On (π 1βπ₯) = V | ||
Theorem | rankwflem 9810* | Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 9787 is useful in proofs of theorems about the rank function. (Contributed by NM, 4-Oct-2003.) |
β’ (π΄ β π β βπ₯ β On π΄ β (π 1βsuc π₯)) | ||
Theorem | rankval 9811* | Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). (Contributed by NM, 24-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
β’ π΄ β V β β’ (rankβπ΄) = β© {π₯ β On β£ π΄ β (π 1βsuc π₯)} | ||
Theorem | rankvalg 9812* | Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9811 expresses the class existence requirement as an antecedent instead of a hypothesis. (Contributed by NM, 5-Oct-2003.) |
β’ (π΄ β π β (rankβπ΄) = β© {π₯ β On β£ π΄ β (π 1βsuc π₯)}) | ||
Theorem | rankval2 9813* | Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. (Contributed by NM, 8-Oct-2003.) |
β’ (π΄ β π΅ β (rankβπ΄) = β© {π₯ β On β£ π΄ β (π 1βπ₯)}) | ||
Theorem | uniwf 9814 | A union is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (π΄ β βͺ (π 1 β On) β βͺ π΄ β βͺ (π 1 β On)) | ||
Theorem | rankr1clem 9815 | Lemma for rankr1c 9816. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β dom π 1) β (Β¬ π΄ β (π 1βπ΅) β π΅ β (rankβπ΄))) | ||
Theorem | rankr1c 9816 | A relationship between the rank function and the cumulative hierarchy of sets function π 1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (π΄ β βͺ (π 1 β On) β (π΅ = (rankβπ΄) β (Β¬ π΄ β (π 1βπ΅) β§ π΄ β (π 1βsuc π΅)))) | ||
Theorem | rankidn 9817 | A relationship between the rank function and the cumulative hierarchy of sets function π 1. (Contributed by Mario Carneiro, 17-Nov-2014.) |
β’ (π΄ β βͺ (π 1 β On) β Β¬ π΄ β (π 1β(rankβπ΄))) | ||
Theorem | rankpwi 9818 | The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 3-Jun-2013.) |
β’ (π΄ β βͺ (π 1 β On) β (rankβπ« π΄) = suc (rankβπ΄)) | ||
Theorem | rankelb 9819 | The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (π΅ β βͺ (π 1 β On) β (π΄ β π΅ β (rankβπ΄) β (rankβπ΅))) | ||
Theorem | wfelirr 9820 | A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 9592. (Contributed by Mario Carneiro, 2-Jan-2017.) |
β’ (π΄ β βͺ (π 1 β On) β Β¬ π΄ β π΄) | ||
Theorem | rankval3b 9821* | The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 17-Nov-2014.) |
β’ (π΄ β βͺ (π 1 β On) β (rankβπ΄) = β© {π₯ β On β£ βπ¦ β π΄ (rankβπ¦) β π₯}) | ||
Theorem | ranksnb 9822 | The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) |
β’ (π΄ β βͺ (π 1 β On) β (rankβ{π΄}) = suc (rankβπ΄)) | ||
Theorem | rankonidlem 9823 | Lemma for rankonid 9824. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.) |
β’ (π΄ β dom π 1 β (π΄ β βͺ (π 1 β On) β§ (rankβπ΄) = π΄)) | ||
Theorem | rankonid 9824 | The rank of an ordinal number is itself. Proposition 9.18 of [TakeutiZaring] p. 79 and its converse. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (π΄ β dom π 1 β (rankβπ΄) = π΄) | ||
Theorem | onwf 9825 | The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ On β βͺ (π 1 β On) | ||
Theorem | onssr1 9826 | Initial segments of the ordinals are contained in initial segments of the cumulative hierarchy. (Contributed by FL, 20-Apr-2011.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (π΄ β dom π 1 β π΄ β (π 1βπ΄)) | ||
Theorem | rankr1g 9827 | A relationship between the rank function and the cumulative hierarchy of sets function π 1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (π΄ β π β (π΅ = (rankβπ΄) β (Β¬ π΄ β (π 1βπ΅) β§ π΄ β (π 1βsuc π΅)))) | ||
Theorem | rankid 9828 | Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V β β’ π΄ β (π 1βsuc (rankβπ΄)) | ||
Theorem | rankr1 9829 | A relationship between the rank function and the cumulative hierarchy of sets function π 1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by NM, 6-Oct-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V β β’ (π΅ = (rankβπ΄) β (Β¬ π΄ β (π 1βπ΅) β§ π΄ β (π 1βsuc π΅))) | ||
Theorem | ssrankr1 9830 | A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets π 1. Proposition 9.15(3) of [TakeutiZaring] p. 79. (Contributed by NM, 8-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V β β’ (π΅ β On β (π΅ β (rankβπ΄) β Β¬ π΄ β (π 1βπ΅))) | ||
Theorem | rankr1a 9831 | A relationship between rank and π 1, clearly equivalent to ssrankr1 9830 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 9859 for the subset version. (Contributed by Raph Levien, 29-May-2004.) |
β’ π΄ β V β β’ (π΅ β On β (π΄ β (π 1βπ΅) β (rankβπ΄) β π΅)) | ||
Theorem | r1val2 9832* | The value of the cumulative hierarchy of sets function expressed in terms of rank. Definition 15.19 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) |
β’ (π΄ β On β (π 1βπ΄) = {π₯ β£ (rankβπ₯) β π΄}) | ||
Theorem | r1val3 9833* | The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (π΄ β On β (π 1βπ΄) = βͺ π₯ β π΄ π« {π¦ β£ (rankβπ¦) β π₯}) | ||
Theorem | rankel 9834 | The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΅ β V β β’ (π΄ β π΅ β (rankβπ΄) β (rankβπ΅)) | ||
Theorem | rankval3 9835* | The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V β β’ (rankβπ΄) = β© {π₯ β On β£ βπ¦ β π΄ (rankβπ¦) β π₯} | ||
Theorem | bndrank 9836* | Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.) |
β’ (βπ₯ β On βπ¦ β π΄ (rankβπ¦) β π₯ β π΄ β V) | ||
Theorem | unbndrank 9837* | The elements of a proper class have unbounded rank. Exercise 2 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.) |
β’ (Β¬ π΄ β V β βπ₯ β On βπ¦ β π΄ π₯ β (rankβπ¦)) | ||
Theorem | rankpw 9838 | The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 22-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V β β’ (rankβπ« π΄) = suc (rankβπ΄) | ||
Theorem | ranklim 9839 | The rank of a set belongs to a limit ordinal iff the rank of its power set does. (Contributed by NM, 18-Sep-2006.) |
β’ (Lim π΅ β ((rankβπ΄) β π΅ β (rankβπ« π΄) β π΅)) | ||
Theorem | r1pw 9840 | A stronger property of π 1 than rankpw 9838. The latter merely proves that π 1 of the successor is a power set, but here we prove that if π΄ is in the cumulative hierarchy, then π« π΄ is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (π΅ β On β (π΄ β (π 1βπ΅) β π« π΄ β (π 1βsuc π΅))) | ||
Theorem | r1pwALT 9841 | Alternate shorter proof of r1pw 9840 based on the additional axioms ax-reg 9587 and ax-inf2 9636. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π΅ β On β (π΄ β (π 1βπ΅) β π« π΄ β (π 1βsuc π΅))) | ||
Theorem | r1pwcl 9842 | The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2014.) |
β’ (Lim π΅ β (π΄ β (π 1βπ΅) β π« π΄ β (π 1βπ΅))) | ||
Theorem | rankssb 9843 | The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (π΅ β βͺ (π 1 β On) β (π΄ β π΅ β (rankβπ΄) β (rankβπ΅))) | ||
Theorem | rankss 9844 | The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΅ β V β β’ (π΄ β π΅ β (rankβπ΄) β (rankβπ΅)) | ||
Theorem | rankunb 9845 | The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β βͺ (π 1 β On)) β (rankβ(π΄ βͺ π΅)) = ((rankβπ΄) βͺ (rankβπ΅))) | ||
Theorem | rankprb 9846 | The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 10-Jun-2013.) |
β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β βͺ (π 1 β On)) β (rankβ{π΄, π΅}) = suc ((rankβπ΄) βͺ (rankβπ΅))) | ||
Theorem | rankopb 9847 | The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by Mario Carneiro, 10-Jun-2013.) |
β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β βͺ (π 1 β On)) β (rankββ¨π΄, π΅β©) = suc suc ((rankβπ΄) βͺ (rankβπ΅))) | ||
Theorem | rankuni2b 9848* | The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 8-Jun-2013.) |
β’ (π΄ β βͺ (π 1 β On) β (rankββͺ π΄) = βͺ π₯ β π΄ (rankβπ₯)) | ||
Theorem | ranksn 9849 | The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V β β’ (rankβ{π΄}) = suc (rankβπ΄) | ||
Theorem | rankuni2 9850* | The rank of a union. Part of Theorem 15.17(iv) of [Monk1] p. 112. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V β β’ (rankββͺ π΄) = βͺ π₯ β π΄ (rankβπ₯) | ||
Theorem | rankun 9851 | The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by NM, 26-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V & β’ π΅ β V β β’ (rankβ(π΄ βͺ π΅)) = ((rankβπ΄) βͺ (rankβπ΅)) | ||
Theorem | rankpr 9852 | The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V & β’ π΅ β V β β’ (rankβ{π΄, π΅}) = suc ((rankβπ΄) βͺ (rankβπ΅)) | ||
Theorem | rankop 9853 | The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 13-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V & β’ π΅ β V β β’ (rankββ¨π΄, π΅β©) = suc suc ((rankβπ΄) βͺ (rankβπ΅)) | ||
Theorem | r1rankid 9854 | Any set is a subset of the hierarchy of its rank. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (π΄ β π β π΄ β (π 1β(rankβπ΄))) | ||
Theorem | rankeq0b 9855 | A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.) |
β’ (π΄ β βͺ (π 1 β On) β (π΄ = β β (rankβπ΄) = β )) | ||
Theorem | rankeq0 9856 | A set is empty iff its rank is empty. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V β β’ (π΄ = β β (rankβπ΄) = β ) | ||
Theorem | rankr1id 9857 | The rank of the hierarchy of an ordinal number is itself. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (π΄ β dom π 1 β (rankβ(π 1βπ΄)) = π΄) | ||
Theorem | rankuni 9858 | The rank of a union. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ (rankββͺ π΄) = βͺ (rankβπ΄) | ||
Theorem | rankr1b 9859 | A relationship between rank and π 1. See rankr1a 9831 for the membership version. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V β β’ (π΅ β On β (π΄ β (π 1βπ΅) β (rankβπ΄) β π΅)) | ||
Theorem | ranksuc 9860 | The rank of a successor. (Contributed by NM, 18-Sep-2006.) |
β’ π΄ β V β β’ (rankβsuc π΄) = suc (rankβπ΄) | ||
Theorem | rankuniss 9861 | Upper bound of the rank of a union. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 30-Nov-2003.) |
β’ π΄ β V β β’ (rankββͺ π΄) β (rankβπ΄) | ||
Theorem | rankval4 9862* | The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. (Contributed by NM, 12-Oct-2003.) |
β’ π΄ β V β β’ (rankβπ΄) = βͺ π₯ β π΄ suc (rankβπ₯) | ||
Theorem | rankbnd 9863* | The rank of a set is bounded by a bound for the successor of its members. (Contributed by NM, 18-Sep-2006.) |
β’ π΄ β V β β’ (βπ₯ β π΄ suc (rankβπ₯) β π΅ β (rankβπ΄) β π΅) | ||
Theorem | rankbnd2 9864* | The rank of a set is bounded by the successor of a bound for its members. (Contributed by NM, 15-Sep-2006.) |
β’ π΄ β V β β’ (π΅ β On β (βπ₯ β π΄ (rankβπ₯) β π΅ β (rankβπ΄) β suc π΅)) | ||
Theorem | rankc1 9865* | A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.) |
β’ π΄ β V β β’ (βπ₯ β π΄ (rankβπ₯) β (rankββͺ π΄) β (rankβπ΄) = (rankββͺ π΄)) | ||
Theorem | rankc2 9866* | A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.) |
β’ π΄ β V β β’ (βπ₯ β π΄ (rankβπ₯) = (rankββͺ π΄) β (rankβπ΄) = suc (rankββͺ π΄)) | ||
Theorem | rankelun 9867 | Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V & β’ π΅ β V & β’ πΆ β V & β’ π· β V β β’ (((rankβπ΄) β (rankβπΆ) β§ (rankβπ΅) β (rankβπ·)) β (rankβ(π΄ βͺ π΅)) β (rankβ(πΆ βͺ π·))) | ||
Theorem | rankelpr 9868 | Rank membership is inherited by unordered pairs. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
β’ π΄ β V & β’ π΅ β V & β’ πΆ β V & β’ π· β V β β’ (((rankβπ΄) β (rankβπΆ) β§ (rankβπ΅) β (rankβπ·)) β (rankβ{π΄, π΅}) β (rankβ{πΆ, π·})) | ||
Theorem | rankelop 9869 | Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.) |
β’ π΄ β V & β’ π΅ β V & β’ πΆ β V & β’ π· β V β β’ (((rankβπ΄) β (rankβπΆ) β§ (rankβπ΅) β (rankβπ·)) β (rankββ¨π΄, π΅β©) β (rankββ¨πΆ, π·β©)) | ||
Theorem | rankxpl 9870 | A lower bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006.) |
β’ π΄ β V & β’ π΅ β V β β’ ((π΄ Γ π΅) β β β (rankβ(π΄ βͺ π΅)) β (rankβ(π΄ Γ π΅))) | ||
Theorem | rankxpu 9871 | An upper bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006.) |
β’ π΄ β V & β’ π΅ β V β β’ (rankβ(π΄ Γ π΅)) β suc suc (rankβ(π΄ βͺ π΅)) | ||
Theorem | rankfu 9872 | An upper bound on the rank of a function. (Contributed by GΓ©rard Lang, 5-Aug-2018.) |
β’ π΄ β V & β’ π΅ β V β β’ (πΉ:π΄βΆπ΅ β (rankβπΉ) β suc suc (rankβ(π΄ βͺ π΅))) | ||
Theorem | rankmapu 9873 | An upper bound on the rank of set exponentiation. (Contributed by GΓ©rard Lang, 5-Aug-2018.) |
β’ π΄ β V & β’ π΅ β V β β’ (rankβ(π΄ βm π΅)) β suc suc suc (rankβ(π΄ βͺ π΅)) | ||
Theorem | rankxplim 9874 | The rank of a Cartesian product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 9877 for the successor case. (Contributed by NM, 19-Sep-2006.) |
β’ π΄ β V & β’ π΅ β V β β’ ((Lim (rankβ(π΄ βͺ π΅)) β§ (π΄ Γ π΅) β β ) β (rankβ(π΄ Γ π΅)) = (rankβ(π΄ βͺ π΅))) | ||
Theorem | rankxplim2 9875 | If the rank of a Cartesian product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.) |
β’ π΄ β V & β’ π΅ β V β β’ (Lim (rankβ(π΄ Γ π΅)) β Lim (rankβ(π΄ βͺ π΅))) | ||
Theorem | rankxplim3 9876 | The rank of a Cartesian product is a limit ordinal iff its union is. (Contributed by NM, 19-Sep-2006.) |
β’ π΄ β V & β’ π΅ β V β β’ (Lim (rankβ(π΄ Γ π΅)) β Lim βͺ (rankβ(π΄ Γ π΅))) | ||
Theorem | rankxpsuc 9877 | The rank of a Cartesian product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 9874 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.) |
β’ π΄ β V & β’ π΅ β V β β’ (((rankβ(π΄ βͺ π΅)) = suc πΆ β§ (π΄ Γ π΅) β β ) β (rankβ(π΄ Γ π΅)) = suc suc (rankβ(π΄ βͺ π΅))) | ||
Theorem | tcwf 9878 | The transitive closure function is well-founded if its argument is. (Contributed by Mario Carneiro, 23-Jun-2013.) |
β’ (π΄ β βͺ (π 1 β On) β (TCβπ΄) β βͺ (π 1 β On)) | ||
Theorem | tcrank 9879 | This theorem expresses two different facts from the two subset implications in this equality. In the forward direction, it says that the transitive closure has members of every rank below π΄. Stated another way, to construct a set at a given rank, you have to climb the entire hierarchy of ordinals below (rankβπ΄), constructing at least one set at each level in order to move up the ranks. In the reverse direction, it says that every member of (TCβπ΄) has a rank below the rank of π΄, since intuitively it contains only the members of π΄ and the members of those and so on, but nothing "bigger" than π΄. (Contributed by Mario Carneiro, 23-Jun-2013.) |
β’ (π΄ β βͺ (π 1 β On) β (rankβπ΄) = (rank β (TCβπ΄))) | ||
Theorem | scottex 9880* | Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set. (Contributed by NM, 13-Oct-2003.) |
β’ {π₯ β π΄ β£ βπ¦ β π΄ (rankβπ₯) β (rankβπ¦)} β V | ||
Theorem | scott0 9881* | Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. π΄ is empty). (Contributed by NM, 15-Oct-2003.) |
β’ (π΄ = β β {π₯ β π΄ β£ βπ¦ β π΄ (rankβπ₯) β (rankβπ¦)} = β ) | ||
Theorem | scottexs 9882* | Theorem scheme version of scottex 9880. The collection of all π₯ of minimum rank such that π(π₯) is true, is a set. (Contributed by NM, 13-Oct-2003.) |
β’ {π₯ β£ (π β§ βπ¦([π¦ / π₯]π β (rankβπ₯) β (rankβπ¦)))} β V | ||
Theorem | scott0s 9883* | Theorem scheme version of scott0 9881. The collection of all π₯ of minimum rank such that π(π₯) is true, is not empty iff there is an π₯ such that π(π₯) holds. (Contributed by NM, 13-Oct-2003.) |
β’ (βπ₯π β {π₯ β£ (π β§ βπ¦([π¦ / π₯]π β (rankβπ₯) β (rankβπ¦)))} β β ) | ||
Theorem | cplem1 9884* | Lemma for the Collection Principle cp 9886. (Contributed by NM, 17-Oct-2003.) |
β’ πΆ = {π¦ β π΅ β£ βπ§ β π΅ (rankβπ¦) β (rankβπ§)} & β’ π· = βͺ π₯ β π΄ πΆ β β’ βπ₯ β π΄ (π΅ β β β (π΅ β© π·) β β ) | ||
Theorem | cplem2 9885* | Lemma for the Collection Principle cp 9886. (Contributed by NM, 17-Oct-2003.) |
β’ π΄ β V β β’ βπ¦βπ₯ β π΄ (π΅ β β β (π΅ β© π¦) β β ) | ||
Theorem | cp 9886* | Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 9880 that collapses a proper class into a set of minimum rank. The wff π can be thought of as π(π₯, π¦). Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.) |
β’ βπ€βπ₯ β π§ (βπ¦π β βπ¦ β π€ π) | ||
Theorem | bnd 9887* | A very strong generalization of the Axiom of Replacement (compare zfrep6 7941), derived from the Collection Principle cp 9886. Its strength lies in the rather profound fact that π(π₯, π¦) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. (Contributed by NM, 17-Oct-2004.) |
β’ (βπ₯ β π§ βπ¦π β βπ€βπ₯ β π§ βπ¦ β π€ π) | ||
Theorem | bnd2 9888* | A variant of the Boundedness Axiom bnd 9887 that picks a subset π§ out of a possibly proper class π΅ in which a property is true. (Contributed by NM, 4-Feb-2004.) |
β’ π΄ β V β β’ (βπ₯ β π΄ βπ¦ β π΅ π β βπ§(π§ β π΅ β§ βπ₯ β π΄ βπ¦ β π§ π)) | ||
Theorem | kardex 9889* | The collection of all sets equinumerous to a set π΄ and having the least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222. (Contributed by NM, 14-Dec-2003.) |
β’ {π₯ β£ (π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦)))} β V | ||
Theorem | karden 9890* | If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 10546). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 9889 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {π₯ β£ π₯ β π΄}. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.) |
β’ π΄ β V & β’ πΆ = {π₯ β£ (π₯ β π΄ β§ βπ¦(π¦ β π΄ β (rankβπ₯) β (rankβπ¦)))} & β’ π· = {π₯ β£ (π₯ β π΅ β§ βπ¦(π¦ β π΅ β (rankβπ₯) β (rankβπ¦)))} β β’ (πΆ = π· β π΄ β π΅) | ||
Theorem | htalem 9891* | Lemma for defining an emulation of Hilbert's epsilon. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem is equivalent to Hilbert's "transfinite axiom", described on that page, with the additional π We π΄ antecedent. The element π΅ is the epsilon that the theorem emulates. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.) |
β’ π΄ β V & β’ π΅ = (β©π₯ β π΄ βπ¦ β π΄ Β¬ π¦π π₯) β β’ ((π We π΄ β§ π΄ β β ) β π΅ β π΄) | ||
Theorem | hta 9892* |
A ZFC emulation of Hilbert's transfinite axiom. The set π΅ has the
properties of Hilbert's epsilon, except that it also depends on a
well-ordering π
. This theorem arose from
discussions with Raph
Levien on 5-Mar-2004 about translating the HOL proof language, which
uses Hilbert's epsilon. See
https://us.metamath.org/downloads/choice.txt
(copy of obsolete link
http://ghilbert.org/choice.txt) and
https://us.metamath.org/downloads/megillaward2005he.pdf.
Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem differs from Hilbert's transfinite axiom described on that page in that it requires π We π΄ as an antecedent. Class π΄ collects the sets of the least rank for which π(π₯) is true. Class π΅, which emulates Hilbert's epsilon, is the minimum element in a well-ordering π on π΄. If a well-ordering π on π΄ can be expressed in a closed form, as might be the case if we are working with say natural numbers, we can eliminate the antecedent with modus ponens, giving us the exact equivalent of Hilbert's transfinite axiom. Otherwise, we replace π with a dummy setvar variable, say π€, and attach π€ We π΄ as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point, π΅ (which will have π€ as a free variable) will no longer be present, and we can eliminate π€ We π΄ by applying exlimiv 1934 and weth 10490, using scottexs 9882 to establish the existence of π΄. For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9891. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.) |
β’ π΄ = {π₯ β£ (π β§ βπ¦([π¦ / π₯]π β (rankβπ₯) β (rankβπ¦)))} & β’ π΅ = (β©π§ β π΄ βπ€ β π΄ Β¬ π€π π§) β β’ (π We π΄ β (π β [π΅ / π₯]π)) | ||
Syntax | cdju 9893 | Extend class notation to include disjoint union of two classes. |
class (π΄ β π΅) | ||
Syntax | cinl 9894 | Extend class notation to include left injection of a disjoint union. |
class inl | ||
Syntax | cinr 9895 | Extend class notation to include right injection of a disjoint union. |
class inr | ||
Definition | df-dju 9896 | Disjoint union of two classes. This is a way of creating a set which contains elements corresponding to each element of π΄ or π΅, tagging each one with whether it came from π΄ or π΅. (Contributed by Jim Kingdon, 20-Jun-2022.) |
β’ (π΄ β π΅) = (({β } Γ π΄) βͺ ({1o} Γ π΅)) | ||
Definition | df-inl 9897 | Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
β’ inl = (π₯ β V β¦ β¨β , π₯β©) | ||
Definition | df-inr 9898 | Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
β’ inr = (π₯ β V β¦ β¨1o, π₯β©) | ||
Theorem | djueq12 9899 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
β’ ((π΄ = π΅ β§ πΆ = π·) β (π΄ β πΆ) = (π΅ β π·)) | ||
Theorem | djueq1 9900 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
β’ (π΄ = π΅ β (π΄ β πΆ) = (π΅ β πΆ)) |
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