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

Theoremaxdclem 8401* Lemma for axdc 8403. (Contributed by Mario Carneiro, 25-Jan-2013.)

Theoremaxdclem2 8402* Lemma for axdc 8403. Using the full Axiom of Choice, we can construct a choice function on . From this, we can build a sequence starting at any value by repeatedly applying to the set (where is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)

Theoremaxdc 8403* This theorem derives ax-dc 8328 using ax-ac 8341 and ax-inf 7595. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (Contributed by Mario Carneiro, 25-Jan-2013.)

Theoremfodom 8404 An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8356. AC is not needed for finite sets - see fodomfi 7387. See also fodomnum 7940. (Contributed by NM, 23-Jul-2004.)

Theoremfodomg 8405 An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)

Theoremfodomb 8406* Equivalence of an onto mapping and dominance for a non-empty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.)

Theoremwdomac 8407 When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
*

Theorembrdom3 8408* Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)

Theorembrdom5 8409* An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)

Theorembrdom4 8410* An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Theorembrdom7disj 8411* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Theorembrdom6disj 8412* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)

Theoremfin71ac 8413 Once we allow AC, the "strongest" definition of finite set becomes equivalent to the "weakest" and the entire hierarchy collapses. (Contributed by Stefan O'Rear, 29-Oct-2014.)
FinVII

Theoremimadomg 8414 An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.)

Theoremfnrndomg 8415 The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)

Theoremiunfo 8416* Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)

Theoremiundom2g 8417* An upper bound for the cardinality of a disjoint indexed union, with explicit choice principles. depends on and should be thought of as . (Contributed by Mario Carneiro, 1-Sep-2015.)
AC

Theoremiundomg 8418* An upper bound for the cardinality of an indexed union, with explicit choice principles. depends on and should be thought of as . (Contributed by Mario Carneiro, 1-Sep-2015.)
AC               AC

Theoremiundom 8419* An upper bound for the cardinality of an indexed union. depends on and should be thought of as . (Contributed by NM, 26-Mar-2006.)

Theoremunidom 8420* An upper bound for the cardinality of a union. Theorem 10.47 of [TakeutiZaring] p. 98. (Contributed by NM, 25-Mar-2006.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Theoremuniimadom 8421* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. (Contributed by NM, 25-Mar-2006.)

Theoremuniimadomf 8422* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8421 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)

3.2.3  Cardinal number theorems using Axiom of Choice

Theoremcardval 8423* The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 7880 for a simpler version of its value. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremcardid 8424 Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremcardidg 8425 Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8424. (Contributed by David Moews, 1-May-2017.)

Theoremcardidd 8426 Any set is equinumerous to its cardinal number. Deduction form of cardid 8424. (Contributed by David Moews, 1-May-2017.)

Theoremcardf 8427 The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremcarden 8428 Two sets are equinumerous iff their cardinal numbers are equal. This important theorem expresses the essential concept behind "cardinality" or "size." This theorem appears as Proposition 10.10 of [TakeutiZaring] p. 85, Theorem 7P of [Enderton] p. 197, and Theorem 9 of [Suppes] p. 242 (among others). The Axiom of Choice is required for its proof.

The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank (see karden 7821). (Contributed by NM, 22-Oct-2003.)

Theoremcardeq0 8429 Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)

Theoremunsnen 8430 Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.)

Theoremcarddom 8431 Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of [Quine] p. 232. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremcardsdom 8432 Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of [Eisenberg] p. 310. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremdomtri 8433 Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theorementric 8434 Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of [Suppes] p. 242. (Contributed by NM, 4-Jan-2004.)

Theorementri2 8435 Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)

Theorementri3 8436 Trichotomy of dominance. This theorem is equivalent to the Axiom of Choice. Part of Proposition 4.42(d) of [Mendelson] p. 275. (Contributed by NM, 4-Jan-2004.)

Theoremsdomsdomcard 8437 A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)

Theoremcanth3 8438 Cantor's theorem in terms of cardinals. This theorem tells us that no matter how largei a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.)

Theoreminfxpidm 8439 The cross product of an infinite set with itself is idempotent. This theorem (which is an AC equivalent) provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This proof follows as a corollary of infxpen 7898. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)

Theoremondomon 8440* The collection of ordinal numbers dominated by a set is an ordinal number. (In general, not all collections of ordinal numbers are ordinal.) Theorem 56 of [Suppes] p. 227. This theorem can be proved (with a longer proof) without the Axiom of Choice; see hartogs 7515. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.)

Theoremcardmin 8441* The smallest ordinal that strictly dominates a set is a cardinal. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 20-Sep-2014.)

Theoremficard 8442 A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminfinf 8443 Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)

Theoremunirnfdomd 8444 The union of the range of a function from an infinite set into the class of finite sets is dominated by its domain. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremkonigthlem 8445* Lemma for konigth 8446. (Contributed by Mario Carneiro, 22-Feb-2013.)

Theoremkonigth 8446* Konig's Theorem. If for all , then , where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with regular unions rather than disjoint unions for convenience, but the version with disjoint unions is clearly a special case of this version. The Axiom of Choice is needed for this proof, but it contains AC as a simple corollary (letting , this theorem says that an infinite cartesian product of nonempty sets is nonempty), so this is an AC equivalent. Theorem 11.26 of [TakeutiZaring] p. 107. (Contributed by Mario Carneiro, 22-Feb-2013.)

Theoremalephsucpw 8447 The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 8557 or gchaleph2 8553.) (Contributed by NM, 27-Aug-2005.)

Theoremaleph1 8448 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 8449* 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 8450 A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 8341. See dominf 8327 for a version proved from ax-cc 8317. (Contributed by NM, 25-Mar-2007.)

3.2.4  Cardinal number arithmetic using Axiom of Choice

Theoremiunctb 8451* The countable union of countable sets is countable (indexed union version of unictb 8452). (Contributed by Mario Carneiro, 18-Jan-2014.)

Theoremunictb 8452* The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 8451 for indexed union version. (Contributed by NM, 26-Mar-2006.)

Theoreminfmap 8453* 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 8454 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 8455 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 8456 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 8457* An alternate representation of a successor aleph. Compare alephsuc 7951 and alephsuc2 7963. Equality can be obtained by taking the of the right-hand side then using alephcard 7953 and carden 8428. (Contributed by NM, 23-Oct-2004.)

Theoremalephexp2 8458* An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8456 (which works if the base is less than or equal to the exponent) and infmap 8453 (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 8459 A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)

Theorempwcfsdom 8460* A corollary of Konig's Theorem konigth 8446. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
har

Theoremcfpwsdom 8461 A corollary of Konig's Theorem konigth 8446. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)

Theoremalephom 8462 From canth2 7262, 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 8446 (in the form of cfpwsdom 8461), 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 8463 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 8464 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)

Theoremnd2 8465 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)

Theoremnd3 8466 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)

Theoremnd4 8467 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)

Theoremaxextnd 8468 A version of the Axiom of Extensionality with no distinct variable conditions. (Contributed by NM, 14-Aug-2003.)

Theoremaxrepndlem1 8469* Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)

Theoremaxrepndlem2 8470 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 8471 A version of the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)

Theoremaxunndlem1 8472* Lemma for the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)

Theoremaxunnd 8473 A version of the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)

Theoremaxpowndlem1 8474 Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)

Theoremaxpowndlem2 8475* 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 8476* 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 8477 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 8478 A version of the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)

Theoremaxregndlem1 8479 Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxregndlem2 8480* 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 8481 A version of the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxinfndlem1 8482* Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)

Theoremaxinfnd 8483 A version of the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)

Theoremaxacndlem1 8484 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxacndlem2 8485 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxacndlem3 8486 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxacndlem4 8487* 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 8488* 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 8489 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 8490* Axiom of Extensionality ax-ext 2419, reproved from conditionless ZFC version and predicate calculus. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndrep 8491* Axiom of Replacement ax-rep 4322, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndun 8492* Axiom of Union ax-un 4703, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndpow 8493* Axiom of Power Sets ax-pow 4379, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4392. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndreg 8494* Axiom of Regularity ax-reg 7562, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndinf 8495* Axiom of Infinity ax-inf 7595, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4383 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)

Theoremzfcndac 8496* Axiom of Choice ax-ac 8341, 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 8497 Extend class notation to include the collection of sets that satisfy the GCH.
GCH

Definitiondf-gch 8498* 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 8499* Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremfingch 8500 A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

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