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Theorem List for Metamath Proof Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremalephsucpw 8401 The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 8511 or gchaleph2 8507.) (Contributed by NM, 27-Aug-2005.)

Theoremaleph1 8402 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.)

Theoremalephval2 8403* 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.)

Theoremdominfac 8404 A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 8295. See dominf 8281 for a version proved from ax-cc 8271. (Contributed by NM, 25-Mar-2007.)

3.2.4  Cardinal number arithmetic using Axiom of Choice

Theoremiunctb 8405* The countable union of countable sets is countable (indexed union version of unictb 8406). (Contributed by Mario Carneiro, 18-Jan-2014.)

Theoremunictb 8406* The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 8405 for indexed union version. (Contributed by NM, 26-Mar-2006.)

Theoreminfmap 8407* 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.)

Theoremalephadd 8408 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.)

Theoremalephmul 8409 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.)

Theoremalephexp1 8410 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.)

Theoremalephsuc3 8411* An alternate representation of a successor aleph. Compare alephsuc 7905 and alephsuc2 7917. Equality can be obtained by taking the of the right-hand side then using alephcard 7907 and carden 8382. (Contributed by NM, 23-Oct-2004.)

Theoremalephexp2 8412* An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8410 (which works if the base is less than or equal to the exponent) and infmap 8407 (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.)

3.2.5  Cofinality using Axiom of Choice

Theoremalephreg 8413 A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)

Theorempwcfsdom 8414* A corollary of Konig's Theorem konigth 8400. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
har

Theoremcfpwsdom 8415 A corollary of Konig's Theorem konigth 8400. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)

Theoremalephom 8416 From canth2 7219, we know that , but we cannot prove that (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement is consistent for any ordinal ). However, we can prove that is not equal to , nor , on cofinality grounds, because by Konig's Theorem konigth 8400 (in the form of cfpwsdom 8415), has uncountable cofinality, which eliminates limit alephs like . (The first limit aleph that is not eliminated is , which has cofinality .) (Contributed by Mario Carneiro, 21-Mar-2013.)

Theoremsmobeth 8417 The beth function is strictly monotone. This function is not strictly the beth function, but rather bethA is the same as , 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.)

3.3  ZFC Axioms with no distinct variable requirements

Theoremnd1 8418 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)

Theoremnd2 8419 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)

Theoremnd3 8420 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)

Theoremnd4 8421 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)

Theoremaxextnd 8422 A version of the Axiom of Extensionality with no distinct variable conditions. (Contributed by NM, 14-Aug-2003.)

Theoremaxrepndlem1 8423* Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)

Theoremaxrepndlem2 8424 Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremaxrepnd 8425 A version of the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)

Theoremaxunndlem1 8426* Lemma for the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)

Theoremaxunnd 8427 A version of the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)

Theoremaxpowndlem1 8428 Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)

Theoremaxpowndlem2 8429* Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremaxpowndlem3 8430* Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.)

Theoremaxpowndlem4 8431 Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)

Theoremaxpownd 8432 A version of the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)

Theoremaxregndlem1 8433 Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxregndlem2 8434* Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)

Theoremaxregnd 8435 A version of the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxinfndlem1 8436* Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)

Theoremaxinfnd 8437 A version of the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)

Theoremaxacndlem1 8438 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxacndlem2 8439 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxacndlem3 8440 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxacndlem4 8441* Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)

Theoremaxacndlem5 8442* Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)

Theoremaxacnd 8443 A version of the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)

Theoremzfcndext 8444* Axiom of Extensionality ax-ext 2385, reproved from conditionless ZFC version and predicate calculus. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndrep 8445* Axiom of Replacement ax-rep 4280, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndun 8446* Axiom of Union ax-un 4660, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndpow 8447* Axiom of Power Sets ax-pow 4337, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4350. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndreg 8448* Axiom of Regularity ax-reg 7516, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndinf 8449* Axiom of Infinity ax-inf 7549, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4341 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)

Theoremzfcndac 8450* Axiom of Choice ax-ac 8295, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (New usage is discouraged.) (Proof modification is discouraged.)

3.4  The Generalized Continuum Hypothesis

Syntaxcgch 8451 Extend class notation to include the collection of sets that satisfy the GCH.
GCH

Definitiondf-gch 8452* Define the collection of "GCH-sets", or sets for which the generalized continuum hypothesis holds. In this language the generalized continuum hypothesis can be expressed as GCH . A set satisfies the generalized continuum hypothesis if it is finite or there is no set strictly between and its powerset in cardinality. The continuum hypothesis is equivalent to GCH. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremelgch 8453* Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremfingch 8454 A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchi 8455 The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchen1 8456 If , and is an infinite GCH-set, then in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchen2 8457 If , and is an infinite GCH-set, then in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchor 8458 If , and is an infinite GCH-set, then either or in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremengch 8459 The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.)
GCH GCH

Theoremgchdomtri 8460 Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 8504. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremfpwwe2cbv 8461* Lemma for fpwwe2 8474. (Contributed by Mario Carneiro, 3-Jun-2015.)

Theoremfpwwe2lem1 8462* Lemma for fpwwe2 8474. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem2 8463* Lemma for fpwwe2 8474. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremfpwwe2lem3 8464* Lemma for fpwwe2 8474. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremfpwwe2lem5 8465* Lemma for fpwwe2 8474. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem6 8466* Lemma for fpwwe2 8474. (Contributed by Mario Carneiro, 18-May-2015.)
OrdIso        OrdIso

Theoremfpwwe2lem7 8467* Lemma for fpwwe2 8474. (Contributed by Mario Carneiro, 18-May-2015.)
OrdIso        OrdIso

Theoremfpwwe2lem8 8468* Lemma for fpwwe2 8474. Show by induction that the two isometries and agree on their common domain. (Contributed by Mario Carneiro, 15-May-2015.)
OrdIso        OrdIso

Theoremfpwwe2lem9 8469* Lemma for fpwwe2 8474. Given two well-orders and of parts of , one is an initial segment of the other. (The hypothesis is in order to break the symmetry of and .) (Contributed by Mario Carneiro, 15-May-2015.)
OrdIso        OrdIso

Theoremfpwwe2lem10 8470* Lemma for fpwwe2 8474. Given two well-orders and of parts of , one is an initial segment of the other. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem11 8471* Lemma for fpwwe2 8474. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem12 8472* Lemma for fpwwe2 8474. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfpwwe2lem13 8473* Lemma for fpwwe2 8474. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfpwwe2 8474* Given any function from well-orderings of subsets of to , there is a unique well-ordered subset which "agrees" with in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 7867. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfpwwecbv 8475* Lemma for fpwwe 8477. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwelem 8476* Lemma for fpwwe 8477. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe 8477* Given any function from the powerset of to , canth2 7219 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset which "agrees" with in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 7867. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremcanth4 8478* An "effective" form of Cantor's theorem canth 6498. For any function from the powerset of to , there are two definable sets and which witness non-injectivity of . Corollary 1.3 of [KanamoriPincus] p. 416. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremcanthnumlem 8479* Lemma for canthnum 8480. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremcanthnum 8480 The set of well-orderable subsets of a set strictly dominates . A stronger form of canth2 7219. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremcanthwelem 8481* Lemma for canthnum 8480. (Contributed by Mario Carneiro, 31-May-2015.)

Theoremcanthwe 8482* The set of well-orders of a set strictly dominates . A stronger form of canth2 7219. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)

Theoremcanthp1lem1 8483 Lemma for canthp1 8485. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremcanthp1lem2 8484* Lemma for canthp1 8485. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremcanthp1 8485 A slightly stronger form of Cantor's theorem: For , . Corollary 1.6 of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfinngch 8486 The exclusion of finite sets from consideration in df-gch 8452 is necessary, because otherwise finite sets larger than a singleton would violate the GCH property. (Contributed by Mario Carneiro, 10-Jun-2015.)

Theoremgchcda1 8487 An infinite GCH-set is idempotent under cardinal successor. (Contributed by Mario Carneiro, 18-May-2015.)
GCH

Theoremgchinf 8488 An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)
GCH

Theorempwfseqlem1 8489* Lemma for pwfseq 8495. Derive a contradiction by diagonalization. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwfseqlem2 8490* Lemma for pwfseq 8495. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theorempwfseqlem3 8491* Lemma for pwfseq 8495. Using the construction from pwfseqlem1 8489, produce a function that maps any well-ordered infinite set to an element outside the set. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwfseqlem4a 8492* Lemma for pwfseqlem4 8493. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theorempwfseqlem4 8493* Lemma for pwfseq 8495. Derive a final contradiction from the function in pwfseqlem3 8491. Applying fpwwe2 8474 to it, we get a certain maximal well-ordered subset , but the defining property contradicts our assumption on , so we are reduced to the case of finite. This too is a contradiction, though, because and its preimage under are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwfseqlem5 8494* Lemma for pwfseq 8495. Although in some ways pwfseqlem4 8493 is the "main" part of the proof, one last aspect which makes up a remark in the original text is by far the hardest part to formalize. The main proof relies on the existence of an injection from the set of finite sequences on an infinite set to . Now this alone would not be difficult to prove; this is mostly the claim of fseqen 7864. However, what is needed for the proof is a canonical injection on these sets, so we have to start from scratch pulling together explicit bijections from the lemmas.

If one attempts such a program, it will mostly go through, but there is one key step which is inherently nonconstructive, namely the proof of infxpen 7852. The resolution is not obvious, but it turns out that reversing an infinite ordinal's Cantor normal form absorbs all the non-leading terms (cnfcom3c 7619), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 7855). (Contributed by Mario Carneiro, 31-May-2015.)

har        OrdIso                      seq𝜔

Theorempwfseq 8495* The powerset of a Dedekind-infinite set does not inject into the set of finite sequences. The proof is due to Halbeisen and Shelah. Proposition 1.7 of [KanamoriPincus] p. 418. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwxpndom2 8496 The powerset of a Dedekind-infinite set does not inject into its cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwxpndom 8497 The powerset of a Dedekind-infinite set does not inject into its cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwcdandom 8498 The powerset of a Dedekind-infinite set does not inject into its cardinal sum with itself. (Contributed by Mario Carneiro, 31-May-2015.)

Theoremgchcdaidm 8499 An infinite GCH-set is idempotent under cardinal sum. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
GCH

Theoremgchxpidm 8500 An infinite GCH-set is idempotent under cardinal product. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
GCH

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