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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ifpim4 41501 | Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
β’ ((π β π) β if-(π, π, Β¬ π)) | ||
Theorem | ifpnim2 41502 | Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.) |
β’ (Β¬ (π β π) β if-(π, Β¬ π, π)) | ||
Theorem | ifpim123g 41503 | Implication of conditional logical operators. The right hand side is basically conjunctive normal form which is useful in proofs. (Contributed by RP, 16-Apr-2020.) |
β’ ((if-(π, π, π) β if-(π, π, π)) β ((((π β Β¬ π) β¨ (π β π)) β§ ((π β π) β¨ (π β π))) β§ (((π β π) β¨ (π β π)) β§ ((Β¬ π β π) β¨ (π β π))))) | ||
Theorem | ifpim1g 41504 | Implication of conditional logical operators. (Contributed by RP, 18-Apr-2020.) |
β’ ((if-(π, π, π) β if-(π, π, π)) β (((π β π) β¨ (π β π)) β§ ((π β π) β¨ (π β π)))) | ||
Theorem | ifp1bi 41505 | Substitute the first element of conditional logical operator. (Contributed by RP, 20-Apr-2020.) |
β’ ((if-(π, π, π) β if-(π, π, π)) β ((((π β π) β¨ (π β π)) β§ ((π β π) β¨ (π β π))) β§ (((π β π) β¨ (π β π)) β§ ((π β π) β¨ (π β π))))) | ||
Theorem | ifpbi1b 41506 | When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020.) |
β’ (if-(π, π, π) β if-(π, π, π)) | ||
Theorem | ifpimimb 41507 | Factor conditional logic operator over implication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
β’ (if-(π, (π β π), (π β π)) β (if-(π, π, π) β if-(π, π, π))) | ||
Theorem | ifpororb 41508 | Factor conditional logic operator over disjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
β’ (if-(π, (π β¨ π), (π β¨ π)) β (if-(π, π, π) β¨ if-(π, π, π))) | ||
Theorem | ifpananb 41509 | Factor conditional logic operator over conjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
β’ (if-(π, (π β§ π), (π β§ π)) β (if-(π, π, π) β§ if-(π, π, π))) | ||
Theorem | ifpnannanb 41510 | Factor conditional logic operator over nand in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
β’ (if-(π, (π βΌ π), (π βΌ π)) β (if-(π, π, π) βΌ if-(π, π, π))) | ||
Theorem | ifpor123g 41511 | Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.) |
β’ ((if-(π, π, π) β¨ if-(π, π, π)) β ((((π β Β¬ π) β¨ (π β¨ π)) β§ ((π β π) β¨ (π β¨ π))) β§ (((π β π) β¨ (π β¨ π)) β§ ((Β¬ π β π) β¨ (π β¨ π))))) | ||
Theorem | ifpimim 41512 | Consequnce of implication. (Contributed by RP, 17-Apr-2020.) |
β’ (if-(π, (π β π), (π β π)) β (if-(π, π, π) β if-(π, π, π))) | ||
Theorem | ifpbibib 41513 | Factor conditional logic operator over biconditional in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
β’ (if-(π, (π β π), (π β π)) β (if-(π, π, π) β if-(π, π, π))) | ||
Theorem | ifpxorxorb 41514 | Factor conditional logic operator over xor in terms 2 and 3. (Contributed by RP, 21-Apr-2020.) |
β’ (if-(π, (π β» π), (π β» π)) β (if-(π, π, π) β» if-(π, π, π))) | ||
Theorem | rp-fakeimass 41515 | A special case where implication appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.) |
β’ ((π β¨ π) β (((π β π) β π) β (π β (π β π)))) | ||
Theorem | rp-fakeanorass 41516 | A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by RP, 26-Feb-2020.) |
β’ ((π β π) β (((π β§ π) β¨ π) β (π β§ (π β¨ π)))) | ||
Theorem | rp-fakeoranass 41517 | A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.) |
β’ ((π β π) β (((π β¨ π) β§ π) β (π β¨ (π β§ π)))) | ||
Theorem | rp-fakeinunass 41518 | A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by RP, 26-Feb-2020.) |
β’ (πΆ β π΄ β ((π΄ β© π΅) βͺ πΆ) = (π΄ β© (π΅ βͺ πΆ))) | ||
Theorem | rp-fakeuninass 41519 | A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.) |
β’ (π΄ β πΆ β ((π΄ βͺ π΅) β© πΆ) = (π΄ βͺ (π΅ β© πΆ))) | ||
Membership in the class of finite sets can be expressed in many ways. | ||
Theorem | rp-isfinite5 41520* | A set is said to be finite if it can be put in one-to-one correspondence with all the natural numbers between 1 and some π β β0. (Contributed by RP, 3-Mar-2020.) |
β’ (π΄ β Fin β βπ β β0 (1...π) β π΄) | ||
Theorem | rp-isfinite6 41521* | A set is said to be finite if it is either empty or it can be put in one-to-one correspondence with all the natural numbers between 1 and some π β β. (Contributed by RP, 10-Mar-2020.) |
β’ (π΄ β Fin β (π΄ = β β¨ βπ β β (1...π) β π΄)) | ||
Theorem | intabssd 41522* | When for each element π¦ there is a subset π΄ which may substituted for π₯ such that π¦ satisfying π implies π₯ satisfies π then the intersection of all π₯ that satisfy π is a subclass the intersection of all π¦ that satisfy π. (Contributed by RP, 17-Oct-2020.) |
β’ (π β π΄ β π) & β’ ((π β§ π₯ = π΄) β (π β π)) & β’ (π β π΄ β π¦) β β’ (π β β© {π₯ β£ π} β β© {π¦ β£ π}) | ||
Theorem | eu0 41523* | There is only one empty set. (Contributed by RP, 1-Oct-2023.) |
β’ (βπ₯ Β¬ π₯ β β β§ β!π₯βπ¦ Β¬ π¦ β π₯) | ||
Theorem | epelon2 41524 | Over the ordinal numbers, one may define the relation π΄ E π΅ iff π΄ β π΅ and one finds that, under this ordering, On is a well-ordered class, see epweon 7700. This is a weak form of epelg 5536 which only requires that we know π΅ to be a set. (Contributed by RP, 27-Sep-2023.) |
β’ ((π΄ β On β§ π΅ β On) β (π΄ E π΅ β π΄ β π΅)) | ||
Theorem | ontric3g 41525* | For all π₯, π¦ β On, one and only one of the following hold: π₯ β π¦, π¦ = π₯, or π¦ β π₯. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.) |
β’ βπ₯ β On βπ¦ β On ((π₯ β π¦ β Β¬ (π¦ = π₯ β¨ π¦ β π₯)) β§ (π¦ = π₯ β Β¬ (π₯ β π¦ β¨ π¦ β π₯)) β§ (π¦ β π₯ β Β¬ (π₯ β π¦ β¨ π¦ = π₯))) | ||
Theorem | dfsucon 41526* | π΄ is called a successor ordinal if it is not a limit ordinal and not the empty set. (Contributed by RP, 11-Nov-2023.) |
β’ ((Ord π΄ β§ Β¬ Lim π΄ β§ π΄ β β ) β βπ₯ β On π΄ = suc π₯) | ||
Theorem | snen1g 41527 | A singleton is equinumerous to ordinal one iff its content is a set. (Contributed by RP, 8-Oct-2023.) |
β’ ({π΄} β 1o β π΄ β V) | ||
Theorem | snen1el 41528 | A singleton is equinumerous to ordinal one if its content is an element of it. (Contributed by RP, 8-Oct-2023.) |
β’ ({π΄} β 1o β π΄ β {π΄}) | ||
Theorem | sn1dom 41529 | A singleton is dominated by ordinal one. (Contributed by RP, 29-Oct-2023.) |
β’ {π΄} βΌ 1o | ||
Theorem | pr2dom 41530 | An unordered pair is dominated by ordinal two. (Contributed by RP, 29-Oct-2023.) |
β’ {π΄, π΅} βΌ 2o | ||
Theorem | tr3dom 41531 | An unordered triple is dominated by ordinal three. (Contributed by RP, 29-Oct-2023.) |
β’ {π΄, π΅, πΆ} βΌ 3o | ||
Theorem | ensucne0 41532 | A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof shortened by SN, 16-Nov-2023.) |
β’ (π΄ β suc π΅ β π΄ β β ) | ||
Theorem | ensucne0OLD 41533 | A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π΄ β suc π΅ β π΄ β β ) | ||
Theorem | dfom6 41534 | Let Ο be defined to be the union of the set of all finite ordinals. (Contributed by RP, 27-Sep-2023.) |
β’ Ο = βͺ (On β© Fin) | ||
Theorem | infordmin 41535 | Ο is the smallest infinite ordinal. (Contributed by RP, 27-Sep-2023.) |
β’ βπ₯ β (On β Fin)Ο β π₯ | ||
Theorem | iscard4 41536 | Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.) |
β’ ((cardβπ΄) = π΄ β π΄ β ran card) | ||
Theorem | minregex 41537* | Given any cardinal number π΄, there exists an argument π₯, which yields the least regular uncountable value of β΅ which is greater to or equal to π΄. This proof uses AC. (Contributed by RP, 23-Nov-2023.) |
β’ (π΄ β (ran card β Ο) β βπ₯ β On π₯ = β© {π¦ β On β£ (β β π¦ β§ π΄ β (β΅βπ¦) β§ (cfβ(β΅βπ¦)) = (β΅βπ¦))}) | ||
Theorem | minregex2 41538* | Given any cardinal number π΄, there exists an argument π₯, which yields the least regular uncountable value of β΅ which dominates π΄. This proof uses AC. (Contributed by RP, 24-Nov-2023.) |
β’ (π΄ β (ran card β Ο) β βπ₯ β On π₯ = β© {π¦ β On β£ (β β π¦ β§ π΄ βΌ (β΅βπ¦) β§ (cfβ(β΅βπ¦)) = (β΅βπ¦))}) | ||
Theorem | iscard5 41539* | Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.) |
β’ ((cardβπ΄) = π΄ β (π΄ β On β§ βπ₯ β π΄ Β¬ π₯ β π΄)) | ||
Theorem | elrncard 41540* | Let us define a cardinal number to be an element π΄ β On such that π΄ is not equipotent with any π₯ β π΄. (Contributed by RP, 1-Oct-2023.) |
β’ (π΄ β ran card β (π΄ β On β§ βπ₯ β π΄ Β¬ π₯ β π΄)) | ||
Theorem | harval3 41541* | (harβπ΄) is the least cardinal that is greater than π΄. (Contributed by RP, 4-Nov-2023.) |
β’ (π΄ β dom card β (harβπ΄) = β© {π₯ β ran card β£ π΄ βΊ π₯}) | ||
Theorem | harval3on 41542* | For any ordinal number π΄ let (harβπ΄) denote the least cardinal that is greater than π΄. (Contributed by RP, 4-Nov-2023.) |
β’ (π΄ β On β (harβπ΄) = β© {π₯ β ran card β£ π΄ βΊ π₯}) | ||
Theorem | omssrncard 41543 | All natural numbers are cardinals. (Contributed by RP, 1-Oct-2023.) |
β’ Ο β ran card | ||
Theorem | 0iscard 41544 | 0 is a cardinal number. (Contributed by RP, 1-Oct-2023.) |
β’ β β ran card | ||
Theorem | 1iscard 41545 | 1 is a cardinal number. (Contributed by RP, 1-Oct-2023.) |
β’ 1o β ran card | ||
Theorem | omiscard 41546 | Ο is a cardinal number. (Contributed by RP, 1-Oct-2023.) |
β’ Ο β ran card | ||
Theorem | sucomisnotcard 41547 | Ο +o 1o is not a cardinal number. (Contributed by RP, 1-Oct-2023.) |
β’ Β¬ (Ο +o 1o) β ran card | ||
Theorem | nna1iscard 41548 | For any natural number, the add one operation is results in a cardinal number. (Contributed by RP, 1-Oct-2023.) |
β’ (π β Ο β (π +o 1o) β ran card) | ||
Theorem | har2o 41549 | The least cardinal greater than 2 is 3. (Contributed by RP, 5-Nov-2023.) |
β’ (harβ2o) = 3o | ||
Theorem | en2pr 41550* | A class is equinumerous to ordinal two iff it is a pair of distinct sets. (Contributed by RP, 11-Oct-2023.) |
β’ (π΄ β 2o β βπ₯βπ¦(π΄ = {π₯, π¦} β§ π₯ β π¦)) | ||
Theorem | pr2cv 41551 | If an unordered pair is equinumerous to ordinal two, then both parts are sets. (Contributed by RP, 8-Oct-2023.) |
β’ ({π΄, π΅} β 2o β (π΄ β V β§ π΅ β V)) | ||
Theorem | pr2el1 41552 | If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.) |
β’ ({π΄, π΅} β 2o β π΄ β {π΄, π΅}) | ||
Theorem | pr2cv1 41553 | If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.) |
β’ ({π΄, π΅} β 2o β π΄ β V) | ||
Theorem | pr2el2 41554 | If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.) |
β’ ({π΄, π΅} β 2o β π΅ β {π΄, π΅}) | ||
Theorem | pr2cv2 41555 | If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.) |
β’ ({π΄, π΅} β 2o β π΅ β V) | ||
Theorem | pren2 41556 | An unordered pair is equinumerous to ordinal two iff both parts are sets not equal to each other. (Contributed by RP, 8-Oct-2023.) |
β’ ({π΄, π΅} β 2o β (π΄ β V β§ π΅ β V β§ π΄ β π΅)) | ||
Theorem | pr2eldif1 41557 | If an unordered pair is equinumerous to ordinal two, then a part is an element of the difference of the pair and the singleton of the other part. (Contributed by RP, 21-Oct-2023.) |
β’ ({π΄, π΅} β 2o β π΄ β ({π΄, π΅} β {π΅})) | ||
Theorem | pr2eldif2 41558 | If an unordered pair is equinumerous to ordinal two, then a part is an element of the difference of the pair and the singleton of the other part. (Contributed by RP, 21-Oct-2023.) |
β’ ({π΄, π΅} β 2o β π΅ β ({π΄, π΅} β {π΄})) | ||
Theorem | pren2d 41559 | A pair of two distinct sets is equinumerous to ordinal two. (Contributed by RP, 21-Oct-2023.) |
β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β π΄ β π΅) β β’ (π β {π΄, π΅} β 2o) | ||
Theorem | aleph1min 41560 | (β΅β1o) is the least uncountable ordinal. (Contributed by RP, 18-Nov-2023.) |
β’ (β΅β1o) = β© {π₯ β On β£ Ο βΊ π₯} | ||
Theorem | alephiso2 41561 | β΅ is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.) |
β’ β΅ Isom E , βΊ (On, {π₯ β ran card β£ Ο β π₯}) | ||
Theorem | alephiso3 41562 | β΅ is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.) |
β’ β΅ Isom E , βΊ (On, (ran card β Ο)) | ||
Theorem | pwelg 41563* | The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.) |
β’ (βπ₯ β π΅ (βͺ π₯ β π΅ β§ π« π₯ β π΅) β (π΄ β π΅ β π« π΄ β π΅)) | ||
Theorem | pwinfig 41564* | The powerclass of an infinite set is an infinite set, and vice-versa. Here π΅ is a class which is closed under both the union and the powerclass operations and which may have infinite sets as members. (Contributed by RP, 21-Mar-2020.) |
β’ (βπ₯ β π΅ (βͺ π₯ β π΅ β§ π« π₯ β π΅) β (π΄ β (π΅ β Fin) β π« π΄ β (π΅ β Fin))) | ||
Theorem | pwinfi2 41565 | The powerclass of an infinite set is an infinite set, and vice-versa. Here π is a weak universe. (Contributed by RP, 21-Mar-2020.) |
β’ (π β WUni β (π΄ β (π β Fin) β π« π΄ β (π β Fin))) | ||
Theorem | pwinfi3 41566 | The powerclass of an infinite set is an infinite set, and vice-versa. Here π is a transitive Tarski universe. (Contributed by RP, 21-Mar-2020.) |
β’ ((π β Tarski β§ Tr π) β (π΄ β (π β Fin) β π« π΄ β (π β Fin))) | ||
Theorem | pwinfi 41567 | The powerclass of an infinite set is an infinite set, and vice-versa. (Contributed by RP, 21-Mar-2020.) |
β’ (π΄ β (V β Fin) β π« π΄ β (V β Fin)) | ||
While there is not yet a definition, the finite intersection property of a class is introduced by fiint 9202 where two textbook definitions are shown to be equivalent. This property is seen often with ordinal numbers (onin 6345, ordelinel 6415), chains of sets ordered by the proper subset relation (sorpssin 7659), various sets in the field of topology (inopn 22170, incld 22316, innei 22398, ... ) and "universal" classes like weak universes (wunin 10583, tskin 10629) and the class of all sets (inex1g 5275). | ||
Theorem | fipjust 41568* | A definition of the finite intersection property of a class based on closure under pairwise intersection of its elements is independent of the dummy variables. (Contributed by RP, 1-Jan-2020.) |
β’ (βπ’ β π΄ βπ£ β π΄ (π’ β© π£) β π΄ β βπ₯ β π΄ βπ¦ β π΄ (π₯ β© π¦) β π΄) | ||
Theorem | cllem0 41569* | The class of all sets with property π(π§) is closed under the binary operation on sets defined in π (π₯, π¦). (Contributed by RP, 3-Jan-2020.) |
β’ π = {π§ β£ π} & β’ π β π & β’ (π§ = π β (π β π)) & β’ (π§ = π₯ β (π β π)) & β’ (π§ = π¦ β (π β π)) & β’ ((π β§ π) β π) β β’ βπ₯ β π βπ¦ β π π β π | ||
Theorem | superficl 41570* | The class of all supersets of a class has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.) |
β’ π΄ = {π§ β£ π΅ β π§} β β’ βπ₯ β π΄ βπ¦ β π΄ (π₯ β© π¦) β π΄ | ||
Theorem | superuncl 41571* | The class of all supersets of a class is closed under binary union. (Contributed by RP, 3-Jan-2020.) |
β’ π΄ = {π§ β£ π΅ β π§} β β’ βπ₯ β π΄ βπ¦ β π΄ (π₯ βͺ π¦) β π΄ | ||
Theorem | ssficl 41572* | The class of all subsets of a class has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.) |
β’ π΄ = {π§ β£ π§ β π΅} β β’ βπ₯ β π΄ βπ¦ β π΄ (π₯ β© π¦) β π΄ | ||
Theorem | ssuncl 41573* | The class of all subsets of a class is closed under binary union. (Contributed by RP, 3-Jan-2020.) |
β’ π΄ = {π§ β£ π§ β π΅} β β’ βπ₯ β π΄ βπ¦ β π΄ (π₯ βͺ π¦) β π΄ | ||
Theorem | ssdifcl 41574* | The class of all subsets of a class is closed under class difference. (Contributed by RP, 3-Jan-2020.) |
β’ π΄ = {π§ β£ π§ β π΅} β β’ βπ₯ β π΄ βπ¦ β π΄ (π₯ β π¦) β π΄ | ||
Theorem | sssymdifcl 41575* | The class of all subsets of a class is closed under symmetric difference. (Contributed by RP, 3-Jan-2020.) |
β’ π΄ = {π§ β£ π§ β π΅} β β’ βπ₯ β π΄ βπ¦ β π΄ ((π₯ β π¦) βͺ (π¦ β π₯)) β π΄ | ||
Theorem | fiinfi 41576* | If two classes have the finite intersection property, then so does their intersection. (Contributed by RP, 1-Jan-2020.) |
β’ (π β βπ₯ β π΄ βπ¦ β π΄ (π₯ β© π¦) β π΄) & β’ (π β βπ₯ β π΅ βπ¦ β π΅ (π₯ β© π¦) β π΅) & β’ (π β πΆ = (π΄ β© π΅)) β β’ (π β βπ₯ β πΆ βπ¦ β πΆ (π₯ β© π¦) β πΆ) | ||
Theorem | rababg 41577 | Condition when restricted class is equal to unrestricted class. (Contributed by RP, 13-Aug-2020.) |
β’ (βπ₯(π β π₯ β π΄) β {π₯ β π΄ β£ π} = {π₯ β£ π}) | ||
Theorem | elinintab 41578* | Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 13-Aug-2020.) |
β’ (π΄ β (π΅ β© β© {π₯ β£ π}) β (π΄ β π΅ β§ βπ₯(π β π΄ β π₯))) | ||
Theorem | elmapintrab 41579* | Two ways to say a set is an element of the intersection of a class of images. (Contributed by RP, 16-Aug-2020.) |
β’ πΆ β V & β’ πΆ β π΅ β β’ (π΄ β π β (π΄ β β© {π€ β π« π΅ β£ βπ₯(π€ = πΆ β§ π)} β ((βπ₯π β π΄ β π΅) β§ βπ₯(π β π΄ β πΆ)))) | ||
Theorem | elinintrab 41580* | Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 14-Aug-2020.) |
β’ (π΄ β π β (π΄ β β© {π€ β π« π΅ β£ βπ₯(π€ = (π΅ β© π₯) β§ π)} β ((βπ₯π β π΄ β π΅) β§ βπ₯(π β π΄ β π₯)))) | ||
Theorem | inintabss 41581* | Upper bound on intersection of class and the intersection of a class. (Contributed by RP, 13-Aug-2020.) |
β’ (π΄ β© β© {π₯ β£ π}) β β© {π€ β π« π΄ β£ βπ₯(π€ = (π΄ β© π₯) β§ π)} | ||
Theorem | inintabd 41582* | Value of the intersection of class with the intersection of a nonempty class. (Contributed by RP, 13-Aug-2020.) |
β’ (π β βπ₯π) β β’ (π β (π΄ β© β© {π₯ β£ π}) = β© {π€ β π« π΄ β£ βπ₯(π€ = (π΄ β© π₯) β§ π)}) | ||
Theorem | xpinintabd 41583* | Value of the intersection of Cartesian product with the intersection of a nonempty class. (Contributed by RP, 12-Aug-2020.) |
β’ (π β βπ₯π) β β’ (π β ((π΄ Γ π΅) β© β© {π₯ β£ π}) = β© {π€ β π« (π΄ Γ π΅) β£ βπ₯(π€ = ((π΄ Γ π΅) β© π₯) β§ π)}) | ||
Theorem | relintabex 41584 | If the intersection of a class is a relation, then the class is nonempty. (Contributed by RP, 12-Aug-2020.) |
β’ (Rel β© {π₯ β£ π} β βπ₯π) | ||
Theorem | elcnvcnvintab 41585* | Two ways of saying a set is an element of the converse of the converse of the intersection of a class. (Contributed by RP, 20-Aug-2020.) |
β’ (π΄ β β‘β‘β© {π₯ β£ π} β (π΄ β (V Γ V) β§ βπ₯(π β π΄ β π₯))) | ||
Theorem | relintab 41586* | Value of the intersection of a class when it is a relation. (Contributed by RP, 12-Aug-2020.) |
β’ (Rel β© {π₯ β£ π} β β© {π₯ β£ π} = β© {π€ β π« (V Γ V) β£ βπ₯(π€ = β‘β‘π₯ β§ π)}) | ||
Theorem | nonrel 41587 | A non-relation is equal to the base class with all ordered pairs removed. (Contributed by RP, 25-Oct-2020.) |
β’ (π΄ β β‘β‘π΄) = (π΄ β (V Γ V)) | ||
Theorem | elnonrel 41588 | Only an ordered pair where not both entries are sets could be an element of the non-relation part of class. (Contributed by RP, 25-Oct-2020.) |
β’ (β¨π, πβ© β (π΄ β β‘β‘π΄) β (β β π΄ β§ Β¬ (π β V β§ π β V))) | ||
Theorem | cnvssb 41589 | Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.) |
β’ (Rel π΄ β (π΄ β π΅ β β‘π΄ β β‘π΅)) | ||
Theorem | relnonrel 41590 | The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.) |
β’ (Rel π΄ β (π΄ β β‘β‘π΄) = β ) | ||
Theorem | cnvnonrel 41591 | The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.) |
β’ β‘(π΄ β β‘β‘π΄) = β | ||
Theorem | brnonrel 41592 | A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.) |
β’ ((π β π β§ π β π) β Β¬ π(π΄ β β‘β‘π΄)π) | ||
Theorem | dmnonrel 41593 | The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
β’ dom (π΄ β β‘β‘π΄) = β | ||
Theorem | rnnonrel 41594 | The range of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
β’ ran (π΄ β β‘β‘π΄) = β | ||
Theorem | resnonrel 41595 | A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
β’ ((π΄ β β‘β‘π΄) βΎ π΅) = β | ||
Theorem | imanonrel 41596 | An image under the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
β’ ((π΄ β β‘β‘π΄) β π΅) = β | ||
Theorem | cononrel1 41597 | Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
β’ ((π΄ β β‘β‘π΄) β π΅) = β | ||
Theorem | cononrel2 41598 | Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
β’ (π΄ β (π΅ β β‘β‘π΅)) = β | ||
See also idssxp 5999 by Thierry Arnoux. | ||
Theorem | elmapintab 41599* | Two ways to say a set is an element of mapped intersection of a class. Here πΉ maps elements of πΆ to elements of β© {π₯ β£ π} or π₯. (Contributed by RP, 19-Aug-2020.) |
β’ (π΄ β π΅ β (π΄ β πΆ β§ (πΉβπ΄) β β© {π₯ β£ π})) & β’ (π΄ β πΈ β (π΄ β πΆ β§ (πΉβπ΄) β π₯)) β β’ (π΄ β π΅ β (π΄ β πΆ β§ βπ₯(π β π΄ β πΈ))) | ||
Theorem | fvnonrel 41600 | The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.) |
β’ ((π΄ β β‘β‘π΄)βπ) = β |
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