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Type | Label | Description |
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Statement | ||
Theorem | dfac2 10001* | Axiom of Choice (first form) of [Enderton] p. 49 corresponds to our Axiom of Choice (in the form of ac3 10332). The proof does not make use of AC, but the Axiom of Regularity is used (by applying dfac2b 10000). (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by AV, 16-Jun-2022.) |
β’ (CHOICE β βπ₯βπ¦βπ§ β π₯ (π§ β β β β!π€ β π§ βπ£ β π¦ (π§ β π£ β§ π€ β π£))) | ||
Theorem | dfac7 10002* | Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and our Axiom of Choice (in the form of ac2 10331). The proof does not depend on AC but does depend on the Axiom of Regularity. (Contributed by Mario Carneiro, 17-May-2015.) |
β’ (CHOICE β βπ₯βπ¦βπ§ β π₯ βπ€ β π§ β!π£ β π§ βπ’ β π¦ (π§ β π’ β§ π£ β π’)) | ||
Theorem | dfac0 10003* | Equivalence of two versions of the Axiom of Choice. The proof uses the Axiom of Regularity. The right-hand side is our original ax-ac 10329. (Contributed by Mario Carneiro, 17-May-2015.) |
β’ (CHOICE β βπ₯βπ¦βπ§βπ€((π§ β π€ β§ π€ β π₯) β βπ£βπ’(βπ‘((π’ β π€ β§ π€ β π‘) β§ (π’ β π‘ β§ π‘ β π¦)) β π’ = π£))) | ||
Theorem | dfac1 10004* | Equivalence of two versions of the Axiom of Choice ax-ac 10329. The proof uses the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by Mario Carneiro, 17-May-2015.) |
β’ (CHOICE β βπ₯βπ¦βπ§βπ€((π§ β π€ β§ π€ β π₯) β βπ₯βπ§(βπ₯((π§ β π€ β§ π€ β π₯) β§ (π§ β π₯ β§ π₯ β π¦)) β π§ = π₯))) | ||
Theorem | dfac8 10005* | A proof of the equivalency of the well-ordering theorem weth 10365 and the axiom of choice ac7 10343. (Contributed by Mario Carneiro, 5-Jan-2013.) |
β’ (CHOICE β βπ₯βπ π We π₯) | ||
Theorem | dfac9 10006* | Equivalence of the axiom of choice with a statement related to ac9 10353; definition AC3 of [Schechter] p. 139. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
β’ (CHOICE β βπ((Fun π β§ β β ran π) β Xπ₯ β dom π(πβπ₯) β β )) | ||
Theorem | dfac10 10007 | Axiom of Choice equivalent: the cardinality function measures every set. (Contributed by Mario Carneiro, 6-May-2015.) |
β’ (CHOICE β dom card = V) | ||
Theorem | dfac10c 10008* | Axiom of Choice equivalent: every set is equinumerous to an ordinal. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
β’ (CHOICE β βπ₯βπ¦ β On π¦ β π₯) | ||
Theorem | dfac10b 10009 | Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 9986). (Contributed by Stefan O'Rear, 17-Jan-2015.) |
β’ (CHOICE β ( β β On) = V) | ||
Theorem | acacni 10010 | A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.) |
β’ ((CHOICE β§ π΄ β π) β AC π΄ = V) | ||
Theorem | dfacacn 10011 | A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.) |
β’ (CHOICE β βπ₯AC π₯ = V) | ||
Theorem | dfac13 10012 | The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.) |
β’ (CHOICE β βπ₯ π₯ β AC π₯) | ||
Theorem | dfac12lem1 10013* | Lemma for dfac12 10019. (Contributed by Mario Carneiro, 29-May-2015.) |
β’ (π β π΄ β On) & β’ (π β πΉ:π« (harβ(π 1βπ΄))β1-1βOn) & β’ πΊ = recs((π₯ β V β¦ (π¦ β (π 1βdom π₯) β¦ if(dom π₯ = βͺ dom π₯, ((suc βͺ ran βͺ ran π₯ Β·o (rankβπ¦)) +o ((π₯βsuc (rankβπ¦))βπ¦)), (πΉβ((β‘OrdIso( E , ran (π₯ββͺ dom π₯)) β (π₯ββͺ dom π₯)) β π¦)))))) & β’ (π β πΆ β On) & β’ π» = (β‘OrdIso( E , ran (πΊββͺ πΆ)) β (πΊββͺ πΆ)) β β’ (π β (πΊβπΆ) = (π¦ β (π 1βπΆ) β¦ if(πΆ = βͺ πΆ, ((suc βͺ ran βͺ (πΊ β πΆ) Β·o (rankβπ¦)) +o ((πΊβsuc (rankβπ¦))βπ¦)), (πΉβ(π» β π¦))))) | ||
Theorem | dfac12lem2 10014* | Lemma for dfac12 10019. (Contributed by Mario Carneiro, 29-May-2015.) |
β’ (π β π΄ β On) & β’ (π β πΉ:π« (harβ(π 1βπ΄))β1-1βOn) & β’ πΊ = recs((π₯ β V β¦ (π¦ β (π 1βdom π₯) β¦ if(dom π₯ = βͺ dom π₯, ((suc βͺ ran βͺ ran π₯ Β·o (rankβπ¦)) +o ((π₯βsuc (rankβπ¦))βπ¦)), (πΉβ((β‘OrdIso( E , ran (π₯ββͺ dom π₯)) β (π₯ββͺ dom π₯)) β π¦)))))) & β’ (π β πΆ β On) & β’ π» = (β‘OrdIso( E , ran (πΊββͺ πΆ)) β (πΊββͺ πΆ)) & β’ (π β πΆ β π΄) & β’ (π β βπ§ β πΆ (πΊβπ§):(π 1βπ§)β1-1βOn) β β’ (π β (πΊβπΆ):(π 1βπΆ)β1-1βOn) | ||
Theorem | dfac12lem3 10015* | Lemma for dfac12 10019. (Contributed by Mario Carneiro, 29-May-2015.) |
β’ (π β π΄ β On) & β’ (π β πΉ:π« (harβ(π 1βπ΄))β1-1βOn) & β’ πΊ = recs((π₯ β V β¦ (π¦ β (π 1βdom π₯) β¦ if(dom π₯ = βͺ dom π₯, ((suc βͺ ran βͺ ran π₯ Β·o (rankβπ¦)) +o ((π₯βsuc (rankβπ¦))βπ¦)), (πΉβ((β‘OrdIso( E , ran (π₯ββͺ dom π₯)) β (π₯ββͺ dom π₯)) β π¦)))))) β β’ (π β (π 1βπ΄) β dom card) | ||
Theorem | dfac12r 10016 | The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 10019 does not assume the Axiom of Regularity. (Contributed by Mario Carneiro, 29-May-2015.) |
β’ (βπ₯ β On π« π₯ β dom card β βͺ (π 1 β On) β dom card) | ||
Theorem | dfac12k 10017* | Equivalence of dfac12 10019 and dfac12a 10018, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.) |
β’ (βπ₯ β On π« π₯ β dom card β βπ¦ β On π« (β΅βπ¦) β dom card) | ||
Theorem | dfac12a 10018 | The axiom of choice holds iff every ordinal has a well-orderable powerset. (Contributed by Mario Carneiro, 29-May-2015.) |
β’ (CHOICE β βπ₯ β On π« π₯ β dom card) | ||
Theorem | dfac12 10019 | The axiom of choice holds iff every aleph has a well-orderable powerset. (Contributed by Mario Carneiro, 21-May-2015.) |
β’ (CHOICE β βπ₯ β On π« (β΅βπ₯) β dom card) | ||
Theorem | kmlem1 10020* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, 1 => 2. (Contributed by NM, 5-Apr-2004.) |
β’ (βπ₯((βπ§ β π₯ π§ β β β§ βπ§ β π₯ βπ€ β π₯ π) β βπ¦βπ§ β π₯ π) β βπ₯(βπ§ β π₯ βπ€ β π₯ π β βπ¦βπ§ β π₯ (π§ β β β π))) | ||
Theorem | kmlem2 10021* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.) |
β’ (βπ¦βπ§ β π₯ (π β β!π€ π€ β (π§ β© π¦)) β βπ¦(Β¬ π¦ β π₯ β§ βπ§ β π₯ (π β β!π€ π€ β (π§ β© π¦)))) | ||
Theorem | kmlem3 10022* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. The right-hand side is part of the hypothesis of 4. (Contributed by NM, 25-Mar-2004.) |
β’ ((π§ β βͺ (π₯ β {π§})) β β β βπ£ β π§ βπ€ β π₯ (π§ β π€ β Β¬ π£ β (π§ β© π€))) | ||
Theorem | kmlem4 10023* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.) |
β’ ((π€ β π₯ β§ π§ β π€) β ((π§ β βͺ (π₯ β {π§})) β© π€) = β ) | ||
Theorem | kmlem5 10024* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.) |
β’ ((π€ β π₯ β§ π§ β π€) β ((π§ β βͺ (π₯ β {π§})) β© (π€ β βͺ (π₯ β {π€}))) = β ) | ||
Theorem | kmlem6 10025* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.) |
β’ ((βπ§ β π₯ π§ β β β§ βπ§ β π₯ βπ€ β π₯ (π β π΄ = β )) β βπ§ β π₯ βπ£ β π§ βπ€ β π₯ (π β Β¬ π£ β π΄)) | ||
Theorem | kmlem7 10026* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.) |
β’ ((βπ§ β π₯ π§ β β β§ βπ§ β π₯ βπ€ β π₯ (π§ β π€ β (π§ β© π€) = β )) β Β¬ βπ§ β π₯ βπ£ β π§ βπ€ β π₯ (π§ β π€ β§ π£ β (π§ β© π€))) | ||
Theorem | kmlem8 10027* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 4-Apr-2004.) |
β’ ((Β¬ βπ§ β π’ βπ€ β π§ π β βπ¦βπ§ β π’ (π§ β β β β!π€ π€ β (π§ β© π¦))) β (βπ§ β π’ βπ€ β π§ π β¨ βπ¦(Β¬ π¦ β π’ β§ βπ§ β π’ β!π€ π€ β (π§ β© π¦)))) | ||
Theorem | kmlem9 10028* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.) |
β’ π΄ = {π’ β£ βπ‘ β π₯ π’ = (π‘ β βͺ (π₯ β {π‘}))} β β’ βπ§ β π΄ βπ€ β π΄ (π§ β π€ β (π§ β© π€) = β ) | ||
Theorem | kmlem10 10029* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.) |
β’ π΄ = {π’ β£ βπ‘ β π₯ π’ = (π‘ β βͺ (π₯ β {π‘}))} β β’ (ββ(βπ§ β β βπ€ β β (π§ β π€ β (π§ β© π€) = β ) β βπ¦βπ§ β β π) β βπ¦βπ§ β π΄ π) | ||
Theorem | kmlem11 10030* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.) |
β’ π΄ = {π’ β£ βπ‘ β π₯ π’ = (π‘ β βͺ (π₯ β {π‘}))} β β’ (π§ β π₯ β (π§ β© βͺ π΄) = (π§ β βͺ (π₯ β {π§}))) | ||
Theorem | kmlem12 10031* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 27-Mar-2004.) |
β’ π΄ = {π’ β£ βπ‘ β π₯ π’ = (π‘ β βͺ (π₯ β {π‘}))} β β’ (βπ§ β π₯ (π§ β βͺ (π₯ β {π§})) β β β (βπ§ β π΄ (π§ β β β β!π£ π£ β (π§ β© π¦)) β βπ§ β π₯ (π§ β β β β!π£ π£ β (π§ β© (π¦ β© βͺ π΄))))) | ||
Theorem | kmlem13 10032* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 5-Apr-2004.) |
β’ π΄ = {π’ β£ βπ‘ β π₯ π’ = (π‘ β βͺ (π₯ β {π‘}))} β β’ (βπ₯((βπ§ β π₯ π§ β β β§ βπ§ β π₯ βπ€ β π₯ (π§ β π€ β (π§ β© π€) = β )) β βπ¦βπ§ β π₯ β!π£ π£ β (π§ β© π¦)) β βπ₯(Β¬ βπ§ β π₯ βπ£ β π§ βπ€ β π₯ (π§ β π€ β§ π£ β (π§ β© π€)) β βπ¦βπ§ β π₯ (π§ β β β β!π£ π£ β (π§ β© π¦)))) | ||
Theorem | kmlem14 10033* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.) |
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¦) β§ ((π’ β π§ β§ π’ β π¦) β π’ = π£)))) & β’ (π β βπ§ β π₯ β!π£ π£ β (π§ β© π¦)) β β’ (βπ§ β π₯ βπ£ β π§ βπ€ β π₯ (π§ β π€ β§ π£ β (π§ β© π€)) β βπ¦βπ§βπ£βπ’(π¦ β π₯ β§ π)) | ||
Theorem | kmlem15 10034* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.) |
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¦) β§ ((π’ β π§ β§ π’ β π¦) β π’ = π£)))) & β’ (π β βπ§ β π₯ β!π£ π£ β (π§ β© π¦)) β β’ ((Β¬ π¦ β π₯ β§ π) β βπ§βπ£βπ’(Β¬ π¦ β π₯ β§ π)) | ||
Theorem | kmlem16 10035* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4 5 <=> 4. (Contributed by NM, 4-Apr-2004.) |
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¦) β§ ((π’ β π§ β§ π’ β π¦) β π’ = π£)))) & β’ (π β βπ§ β π₯ β!π£ π£ β (π§ β© π¦)) β β’ ((βπ§ β π₯ βπ£ β π§ βπ€ β π₯ (π§ β π€ β§ π£ β (π§ β© π€)) β¨ βπ¦(Β¬ π¦ β π₯ β§ π)) β βπ¦βπ§βπ£βπ’((π¦ β π₯ β§ π) β¨ (Β¬ π¦ β π₯ β§ π))) | ||
Theorem | dfackm 10036* | Equivalence of the Axiom of Choice and Maes' AC ackm 10335. The proof consists of lemmas kmlem1 10020 through kmlem16 10035 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e., replacing dfac5 9998 with biid 261) establishes the AC equivalence shown by Maes' writeup. The left-hand-side AC shown here was chosen because it is shorter to display. (Contributed by NM, 13-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) |
β’ (CHOICE β βπ₯βπ¦βπ§βπ£βπ’((π¦ β π₯ β§ (π§ β π¦ β ((π£ β π₯ β§ Β¬ π¦ = π£) β§ 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 π¦) β π’ = π£)))))) | ||
For cardinal arithmetic, we follow [Mendelson] p. 258. Rather than defining operations restricted to cardinal numbers, we use disjoint union df-dju 9771 (β) for cardinal addition, Cartesian product df-xp 5637 (Γ) for cardinal multiplication, and set exponentiation df-map 8701 (βm) for cardinal exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 10421, carddom 10424, and cardsdom 10425. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available. | ||
Theorem | undjudom 10037 | Cardinal addition dominates union. (Contributed by NM, 28-Sep-2004.) (Revised by Jim Kingdon, 15-Aug-2023.) |
β’ ((π΄ β π β§ π΅ β π) β (π΄ βͺ π΅) βΌ (π΄ β π΅)) | ||
Theorem | endjudisj 10038 | Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by NM, 5-Apr-2007.) |
β’ ((π΄ β π β§ π΅ β π β§ (π΄ β© π΅) = β ) β (π΄ β π΅) β (π΄ βͺ π΅)) | ||
Theorem | djuen 10039 | Disjoint unions of equinumerous sets are equinumerous. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β π΅ β§ πΆ β π·) β (π΄ β πΆ) β (π΅ β π·)) | ||
Theorem | djuenun 10040 | Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.) |
β’ ((π΄ β π΅ β§ πΆ β π· β§ (π΅ β© π·) = β ) β (π΄ β πΆ) β (π΅ βͺ π·)) | ||
Theorem | dju1en 10041 | Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β π β§ Β¬ π΄ β π΄) β (π΄ β 1o) β suc π΄) | ||
Theorem | dju1dif 10042 | Adding and subtracting one gives back the original cardinality. Similar to pncan 11341 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by Jim Kingdon, 20-Aug-2023.) |
β’ ((π΄ β π β§ π΅ β (π΄ β 1o)) β ((π΄ β 1o) β {π΅}) β π΄) | ||
Theorem | dju1p1e2 10043 | 1+1=2 for cardinal number addition, derived from pm54.43 9871 as promised. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 9765), but after applying definitions, our theorem is equivalent. Because we use a disjoint union for cardinal addition (as explained in the comment at the top of this section), we use β instead of =. See dju1p1e2ALT 10044 for a shorter proof that doesn't use pm54.43 9871. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.) |
β’ (1o β 1o) β 2o | ||
Theorem | dju1p1e2ALT 10044 | Alternate proof of dju1p1e2 10043. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (1o β 1o) β 2o | ||
Theorem | dju0en 10045 | Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ (π΄ β π β (π΄ β β ) β π΄) | ||
Theorem | xp2dju 10046 | Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ (2o Γ π΄) = (π΄ β π΄) | ||
Theorem | djucomen 10047 | Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β π β§ π΅ β π) β (π΄ β π΅) β (π΅ β π΄)) | ||
Theorem | djuassen 10048 | Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β π β§ π΅ β π β§ πΆ β π) β ((π΄ β π΅) β πΆ) β (π΄ β (π΅ β πΆ))) | ||
Theorem | xpdjuen 10049 | Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β π β§ π΅ β π β§ πΆ β π) β (π΄ Γ (π΅ β πΆ)) β ((π΄ Γ π΅) β (π΄ Γ πΆ))) | ||
Theorem | mapdjuen 10050 | Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β π β§ π΅ β π β§ πΆ β π) β (π΄ βm (π΅ β πΆ)) β ((π΄ βm π΅) Γ (π΄ βm πΆ))) | ||
Theorem | pwdjuen 10051 | Sum of exponents law for cardinal arithmetic. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ ((π΄ β π β§ π΅ β π) β π« (π΄ β π΅) β (π« π΄ Γ π« π΅)) | ||
Theorem | djudom1 10052 | Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 1-Sep-2023.) |
β’ ((π΄ βΌ π΅ β§ πΆ β π) β (π΄ β πΆ) βΌ (π΅ β πΆ)) | ||
Theorem | djudom2 10053 | Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ βΌ π΅ β§ πΆ β π) β (πΆ β π΄) βΌ (πΆ β π΅)) | ||
Theorem | djudoml 10054 | A set is dominated by its disjoint union with another. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β π β§ π΅ β π) β π΄ βΌ (π΄ β π΅)) | ||
Theorem | djuxpdom 10055 | Cartesian product dominates disjoint union for sets with cardinality greater than 1. Similar to Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 18-May-2015.) |
β’ ((1o βΊ π΄ β§ 1o βΊ π΅) β (π΄ β π΅) βΌ (π΄ Γ π΅)) | ||
Theorem | djufi 10056 | The disjoint union of two finite sets is finite. (Contributed by NM, 22-Oct-2004.) |
β’ ((π΄ βΊ Ο β§ π΅ βΊ Ο) β (π΄ β π΅) βΊ Ο) | ||
Theorem | cdainflem 10057 | Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.) |
β’ ((π΄ βͺ π΅) β Ο β (π΄ β Ο β¨ π΅ β Ο)) | ||
Theorem | djuinf 10058 | A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ (Ο βΌ π΄ β Ο βΌ (π΄ β π΄)) | ||
Theorem | infdju1 10059 | An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (Ο βΌ π΄ β (π΄ β 1o) β π΄) | ||
Theorem | pwdju1 10060 | The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π΄ β π β (π« π΄ β π« π΄) β π« (π΄ β 1o)) | ||
Theorem | pwdjuidm 10061 | If the natural numbers inject into π΄, then π« π΄ is idempotent under cardinal sum. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (Ο βΌ π΄ β (π« π΄ β π« π΄) β π« π΄) | ||
Theorem | djulepw 10062 | If π΄ is idempotent under cardinal sum and π΅ is dominated by the power set of π΄, then so is the cardinal sum of π΄ and π΅. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (((π΄ β π΄) β π΄ β§ π΅ βΌ π« π΄) β (π΄ β π΅) βΌ π« π΄) | ||
Theorem | onadju 10063 | The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Jim Kingdon, 7-Sep-2023.) |
β’ ((π΄ β On β§ π΅ β On) β (π΄ +o π΅) β (π΄ β π΅)) | ||
Theorem | cardadju 10064 | The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
β’ ((π΄ β dom card β§ π΅ β dom card) β (π΄ β π΅) β ((cardβπ΄) +o (cardβπ΅))) | ||
Theorem | djunum 10065 | The disjoint union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ π΅ β dom card) β (π΄ β π΅) β dom card) | ||
Theorem | unnum 10066 | The union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ π΅ β dom card) β (π΄ βͺ π΅) β dom card) | ||
Theorem | nnadju 10067 | The cardinal and ordinal sums of finite ordinals are equal. For a shorter proof using ax-rep 5241, see nnadjuALT 10068. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.) Avoid ax-rep 5241. (Revised by BTernaryTau, 2-Jul-2024.) |
β’ ((π΄ β Ο β§ π΅ β Ο) β (cardβ(π΄ β π΅)) = (π΄ +o π΅)) | ||
Theorem | nnadjuALT 10068 | Shorter proof of nnadju 10067 using ax-rep 5241. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((π΄ β Ο β§ π΅ β Ο) β (cardβ(π΄ β π΅)) = (π΄ +o π΅)) | ||
Theorem | ficardadju 10069 | The disjoint union of finite sets is equinumerous to the ordinal sum of the cardinalities of those sets. (Contributed by BTernaryTau, 3-Jul-2024.) |
β’ ((π΄ β Fin β§ π΅ β Fin) β (π΄ β π΅) β ((cardβπ΄) +o (cardβπ΅))) | ||
Theorem | ficardun 10070 | The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) Avoid ax-rep 5241. (Revised by BTernaryTau, 3-Jul-2024.) |
β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (cardβ(π΄ βͺ π΅)) = ((cardβπ΄) +o (cardβπ΅))) | ||
Theorem | ficardunOLD 10071 | Obsolete version of ficardun 10070 as of 3-Jul-2024. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((π΄ β Fin β§ π΅ β Fin β§ (π΄ β© π΅) = β ) β (cardβ(π΄ βͺ π΅)) = ((cardβπ΄) +o (cardβπ΅))) | ||
Theorem | ficardun2 10072 | The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.) Avoid ax-rep 5241. (Revised by BTernaryTau, 3-Jul-2024.) |
β’ ((π΄ β Fin β§ π΅ β Fin) β (cardβ(π΄ βͺ π΅)) β ((cardβπ΄) +o (cardβπ΅))) | ||
Theorem | ficardun2OLD 10073 | Obsolete version of ficardun2 10072 as of 3-Jul-2024. (Contributed by Mario Carneiro, 5-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((π΄ β Fin β§ π΅ β Fin) β (cardβ(π΄ βͺ π΅)) β ((cardβπ΄) +o (cardβπ΅))) | ||
Theorem | pwsdompw 10074* | Lemma for domtriom 10313. This is the equinumerosity version of the algebraic identity Ξ£π β π(2βπ) = (2βπ) β 1. (Contributed by Mario Carneiro, 7-Feb-2013.) |
β’ ((π β Ο β§ βπ β suc π(π΅βπ) β π« π) β βͺ π β π (π΅βπ) βΊ (π΅βπ)) | ||
Theorem | unctb 10075 | The union of two countable sets is countable. (Contributed by FL, 25-Aug-2006.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) |
β’ ((π΄ βΌ Ο β§ π΅ βΌ Ο) β (π΄ βͺ π΅) βΌ Ο) | ||
Theorem | infdjuabs 10076 | Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ Ο βΌ π΄ β§ π΅ βΌ π΄) β (π΄ β π΅) β π΄) | ||
Theorem | infunabs 10077 | An infinite set is equinumerous to its union with a smaller one. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ Ο βΌ π΄ β§ π΅ βΌ π΄) β (π΄ βͺ π΅) β π΄) | ||
Theorem | infdju 10078 | The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ π΅ β dom card β§ Ο βΌ π΄) β (π΄ β π΅) β (π΄ βͺ π΅)) | ||
Theorem | infdif 10079 | The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ Ο βΌ π΄ β§ π΅ βΊ π΄) β (π΄ β π΅) β π΄) | ||
Theorem | infdif2 10080 | Cardinality ordering for an infinite class difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ π΅ β dom card β§ Ο βΌ π΄) β ((π΄ β π΅) βΌ π΅ β π΄ βΌ π΅)) | ||
Theorem | infxpdom 10081 | Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β dom card β§ Ο βΌ π΄ β§ π΅ βΌ π΄) β (π΄ Γ π΅) βΌ π΄) | ||
Theorem | infxpabs 10082 | Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ (((π΄ β dom card β§ Ο βΌ π΄) β§ (π΅ β β β§ π΅ βΌ π΄)) β (π΄ Γ π΅) β π΄) | ||
Theorem | infunsdom1 10083 | The union of two sets that are strictly dominated by the infinite set π is also dominated by π. This version of infunsdom 10084 assumes additionally that π΄ is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
β’ (((π β dom card β§ Ο βΌ π) β§ (π΄ βΌ π΅ β§ π΅ βΊ π)) β (π΄ βͺ π΅) βΊ π) | ||
Theorem | infunsdom 10084 | The union of two sets that are strictly dominated by the infinite set π is also strictly dominated by π. (Contributed by Mario Carneiro, 3-May-2015.) |
β’ (((π β dom card β§ Ο βΌ π) β§ (π΄ βΊ π β§ π΅ βΊ π)) β (π΄ βͺ π΅) βΊ π) | ||
Theorem | infxp 10085 | Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ (((π΄ β dom card β§ Ο βΌ π΄) β§ (π΅ β dom card β§ π΅ β β )) β (π΄ Γ π΅) β (π΄ βͺ π΅)) | ||
Theorem | pwdjudom 10086 | A property of dominance over a powerset, and a main lemma for gchac 10551. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.) |
β’ (π« (π΄ β π΄) βΌ (π΄ β π΅) β π« π΄ βΌ π΅) | ||
Theorem | infpss 10087* | Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 10183. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ (Ο βΌ π΄ β βπ₯(π₯ β π΄ β§ π₯ β π΄)) | ||
Theorem | infmap2 10088* | An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 10446 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
β’ ((Ο βΌ π΄ β§ π΅ βΌ π΄ β§ (π΄ βm π΅) β dom card) β (π΄ βm π΅) β {π₯ β£ (π₯ β π΄ β§ π₯ β π΅)}) | ||
Theorem | ackbij2lem1 10089 | Lemma for ackbij2 10113. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
β’ (π΄ β Ο β π« π΄ β (π« Ο β© Fin)) | ||
Theorem | ackbij1lem1 10090 | Lemma for ackbij2 10113. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
β’ (Β¬ π΄ β π΅ β (π΅ β© suc π΄) = (π΅ β© π΄)) | ||
Theorem | ackbij1lem2 10091 | Lemma for ackbij2 10113. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
β’ (π΄ β π΅ β (π΅ β© suc π΄) = ({π΄} βͺ (π΅ β© π΄))) | ||
Theorem | ackbij1lem3 10092 | Lemma for ackbij2 10113. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
β’ (π΄ β Ο β π΄ β (π« Ο β© Fin)) | ||
Theorem | ackbij1lem4 10093 | Lemma for ackbij2 10113. (Contributed by Stefan O'Rear, 19-Nov-2014.) |
β’ (π΄ β Ο β {π΄} β (π« Ο β© Fin)) | ||
Theorem | ackbij1lem5 10094 | Lemma for ackbij2 10113. (Contributed by Stefan O'Rear, 19-Nov-2014.) (Proof shortened by AV, 18-Jul-2022.) |
β’ (π΄ β Ο β (cardβπ« suc π΄) = ((cardβπ« π΄) +o (cardβπ« π΄))) | ||
Theorem | ackbij1lem6 10095 | Lemma for ackbij2 10113. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
β’ ((π΄ β (π« Ο β© Fin) β§ π΅ β (π« Ο β© Fin)) β (π΄ βͺ π΅) β (π« Ο β© Fin)) | ||
Theorem | ackbij1lem7 10096* | Lemma for ackbij1 10108. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
β’ πΉ = (π₯ β (π« Ο β© Fin) β¦ (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦))) β β’ (π΄ β (π« Ο β© Fin) β (πΉβπ΄) = (cardββͺ π¦ β π΄ ({π¦} Γ π« π¦))) | ||
Theorem | ackbij1lem8 10097* | Lemma for ackbij1 10108. (Contributed by Stefan O'Rear, 19-Nov-2014.) |
β’ πΉ = (π₯ β (π« Ο β© Fin) β¦ (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦))) β β’ (π΄ β Ο β (πΉβ{π΄}) = (cardβπ« π΄)) | ||
Theorem | ackbij1lem9 10098* | Lemma for ackbij1 10108. (Contributed by Stefan O'Rear, 19-Nov-2014.) |
β’ πΉ = (π₯ β (π« Ο β© Fin) β¦ (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦))) β β’ ((π΄ β (π« Ο β© Fin) β§ π΅ β (π« Ο β© Fin) β§ (π΄ β© π΅) = β ) β (πΉβ(π΄ βͺ π΅)) = ((πΉβπ΄) +o (πΉβπ΅))) | ||
Theorem | ackbij1lem10 10099* | Lemma for ackbij1 10108. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
β’ πΉ = (π₯ β (π« Ο β© Fin) β¦ (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦))) β β’ πΉ:(π« Ο β© Fin)βΆΟ | ||
Theorem | ackbij1lem11 10100* | Lemma for ackbij1 10108. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
β’ πΉ = (π₯ β (π« Ο β© Fin) β¦ (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦))) β β’ ((π΄ β (π« Ο β© Fin) β§ π΅ β π΄) β π΅ β (π« Ο β© Fin)) |
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