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Theorem List for Metamath Proof Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwdomnumr 7901 Weak dominance agrees with normal for numerable right sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
*

Theoremalephfnon 7902 The aleph function is a function on the class of ordinal numbers. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremaleph0 7903 The first infinite cardinal number, discovered by Georg Cantor in 1873, has the same size as the set of natural numbers (and under our particular definition is also equal to it). In the literature, the argument of the aleph function is often written as a subscript, and the first aleph is written _0. Exercise 3 of [TakeutiZaring] p. 91. Also Definition 12(i) of [Suppes] p. 228. From Moshé Machover, Set Theory, Logic, and Their Limitations, p. 95: "Aleph...the first letter in the Hebrew alphabet...is also the first letter of the Hebrew word...(einsoph, meaning infinity), which is a cabbalistic appellation of the deity. The notation is due to Cantor, who was deeply interested in mysticism." (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremalephlim 7904* Value of the aleph function at a limit ordinal. Definition 12(iii) of [Suppes] p. 91. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremalephsuc 7905 Value of the aleph function at a successor ordinal. Definition 12(ii) of [Suppes] p. 91. Here we express the successor aleph in terms of the Hartogs function df-har 7482, which gives the smallest ordinal that strictly dominates its argument (or the supremum of all ordinals that are dominated by the argument). (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 15-May-2015.)
har

Theoremalephon 7906 An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremalephcard 7907 Every aleph is a cardinal number. Theorem 65 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Theoremalephnbtwn 7908 No cardinal can be sandwiched between an aleph and its successor aleph. Theorem 67 of [Suppes] p. 229. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 15-May-2015.)

Theoremalephnbtwn2 7909 No set has equinumerosity between an aleph and its successor aleph. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Theoremalephordilem1 7910 Lemma for alephordi 7911. (Contributed by NM, 23-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)

Theoremalephordi 7911 Strict ordering property of the aleph function. (Contributed by Mario Carneiro, 2-Feb-2013.)

Theoremalephord 7912 Ordering property of the aleph function. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 9-Feb-2013.)

Theoremalephord2 7913 Ordering property of the aleph function. Theorem 8A(a) of [Enderton] p. 213 and its converse. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 9-Feb-2013.)

Theoremalephord2i 7914 Ordering property of the aleph function. Theorem 66 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.)

Theoremalephord3 7915 Ordering property of the aleph function. (Contributed by NM, 11-Nov-2003.)

Theoremalephsucdom 7916 A set dominated by an aleph is strictly dominated by its successor aleph and vice-versa. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Theoremalephsuc2 7917* An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 7469 function by transfinite recursion, starting from . Using this theorem we could define the aleph function with in place of in df-aleph 7783. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Theoremalephdom 7918 Relationship between inclusion of ordinal numbers and dominance of infinite initial ordinals. (Contributed by Jeff Hankins, 23-Oct-2009.)

Theoremalephgeom 7919 Every aleph is greater than or equal to the set of natural numbers. (Contributed by NM, 11-Nov-2003.)

Theoremalephislim 7920 Every aleph is a limit ordinal. (Contributed by NM, 11-Nov-2003.)

Theoremaleph11 7921 The aleph function is one-to-one. (Contributed by NM, 3-Aug-2004.)

Theoremalephf1 7922 The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. See also alephf1ALT 7940. (Contributed by Mario Carneiro, 2-Feb-2013.)

Theoremalephsdom 7923 If an ordinal is smaller than an initial ordinal, it is strictly dominated by it. (Contributed by Jeff Hankins, 24-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Theoremalephdom2 7924 A dominated initial ordinal is included. (Contributed by Jeff Hankins, 24-Oct-2009.)

Theoremalephle 7925 The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 7946, we will that equality can sometimes hold.) (Contributed by NM, 9-Nov-2003.) (Proof shortened by Mario Carneiro, 22-Feb-2013.)

Theoremcardaleph 7926* Given any transfinite cardinal number , there is exactly one aleph that is equal to it. Here we compute that aleph explicitly. (Contributed by NM, 9-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Theoremcardalephex 7927* Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse. (Contributed by NM, 5-Nov-2003.)

Theoreminfenaleph 7928* An infinite numerable set is equinumerous to an infinite initial ordinal. (Contributed by Jeff Hankins, 23-Oct-2009.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremisinfcard 7929 Two ways to express the property of being a transfinite cardinal. (Contributed by NM, 9-Nov-2003.)

Theoremiscard3 7930 Two ways to express the property of being a cardinal number. (Contributed by NM, 9-Nov-2003.)

Theoremcardnum 7931 Two ways to express the class of all cardinal numbers, which consists of the finite ordinals in plus the transfinite alephs. (Contributed by NM, 10-Sep-2004.)

Theoremalephinit 7932* An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Theoremcarduniima 7933 The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)

Theoremcardinfima 7934* If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)

Theoremalephiso 7935 Aleph is an order isomorphism of the class of ordinal numbers onto the class of infinite cardinals. Definition 10.27 of [TakeutiZaring] p. 90. (Contributed by NM, 3-Aug-2004.)

Theoremalephprc 7936 The class of all transfinite cardinal numbers (the range of the aleph function) is a proper class. Proposition 10.26 of [TakeutiZaring] p. 90. (Contributed by NM, 11-Nov-2003.)

Theoremalephsson 7937 The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)

Theoremunialeph 7938 The union of the class of transfinite cardinals (the range of the aleph function) is the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)

Theoremalephsmo 7939 The aleph function is strictly monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)

Theoremalephf1ALT 7940 The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. (Contributed by Mario Carneiro, 15-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremalephfplem1 7941 Lemma for alephfp 7945. (Contributed by NM, 6-Nov-2004.)

Theoremalephfplem2 7942* Lemma for alephfp 7945. (Contributed by NM, 6-Nov-2004.)

Theoremalephfplem3 7943* Lemma for alephfp 7945. (Contributed by NM, 6-Nov-2004.)

Theoremalephfplem4 7944 Lemma for alephfp 7945. (Contributed by NM, 5-Nov-2004.)

Theoremalephfp 7945 The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 7946 for an abbreviated version just showing existence. (Contributed by NM, 6-Nov-2004.) (Proof shortened by Mario Carneiro, 15-May-2015.)

Theoremalephfp2 7946 The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 7945 for an actual example of a fixed point. Compare the inequality alephle 7925 that holds in general. Note that if is a fixed point, then ... . (Contributed by NM, 6-Nov-2004.) (Revised by Mario Carneiro, 15-May-2015.)

Theoremalephval3 7947* An alternate way to express the value of the aleph function: it is the least infinite cardinal different from all values at smaller arguments. Definition of aleph in [Enderton] p. 212 and definition of aleph in [BellMachover] p. 490 . (Contributed by NM, 16-Nov-2003.)

Theoremalephsucpw2 7948 The power set of an aleph is not strictly dominated by the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 8511 or gchaleph2 8507.) The transposed form alephsucpw 8401 cannot be proven without the AC, and is in fact equlvalent to it. (Contributed by Mario Carneiro, 2-Feb-2013.)

Theoremmappwen 7949 Power rule for cardinal arithmetic. Theorem 11.21 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)

Theoremfinnisoeu 7950* A finite totally ordered set has a unique order isomorphism to a finite ordinal. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 26-Jun-2015.)

Theoremiunfictbso 7951 Countability of a countable union of finite sets with a strict (not globally well) order fulfilling the choice role. (Contributed by Stefan O'Rear, 16-Nov-2014.)

2.6.8  Axiom of Choice equivalents

Syntaxwac 7952 Wff for an abbreviation of the axiom of choice.
CHOICE

Definitiondf-ac 7953* The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49.

There is a slight problem with taking the exact form of ax-ac 8295 as our definition, because the equivalence to more standard forms (dfac2 7967) requires the Axiom of Regularity, which we often try to avoid. Thus, we take the first of the "textbook forms" as the definition and derive the form of ax-ac 8295 itself as dfac0 7969. (Contributed by Mario Carneiro, 22-Feb-2015.)

CHOICE

Theoremaceq1 7954* Equivalence of two versions of the Axiom of Choice ax-ac 8295. The proof uses neither AC nor the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by NM, 5-Apr-2004.)

Theoremaceq0 7955* Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac 8295. (Contributed by NM, 5-Apr-2004.)

Theoremaceq2 7956* Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. (Contributed by NM, 5-Apr-2004.)

Theoremaceq3lem 7957* Lemma for dfac3 7958. (Contributed by NM, 2-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremdfac3 7958* Equivalence of two versions of the Axiom of Choice. The left-hand side is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is the Axiom of Choice of [TakeutiZaring] p. 83. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Stefan O'Rear, 22-Feb-2015.)
CHOICE

Theoremdfac4 7959* Equivalence of two versions of the Axiom of Choice. The right-hand side is Axiom AC of [BellMachover] p. 488. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
CHOICE

Theoremdfac5lem1 7960* Lemma for dfac5 7965. (Contributed by NM, 12-Apr-2004.)

Theoremdfac5lem2 7961* Lemma for dfac5 7965. (Contributed by NM, 12-Apr-2004.)

Theoremdfac5lem3 7962* Lemma for dfac5 7965. (Contributed by NM, 12-Apr-2004.)

Theoremdfac5lem4 7963* Lemma for dfac5 7965. (Contributed by NM, 11-Apr-2004.)

Theoremdfac5lem5 7964* Lemma for dfac5 7965. (Contributed by NM, 12-Apr-2004.)

Theoremdfac5 7965* Equivalence of two versions of the Axiom of Choice. The right-hand side is Theorem 6M(4) of [Enderton] p. 151 and asserts that given a family of mutually disjoint nonempty sets, a set exists containing exactly one member from each set in the family. The proof does not depend on AC. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
CHOICE

Theoremdfac2a 7966* Our Axiom of Choice (in the form of ac3 8298) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2 7967 for the converse (which does use the Axiom of Regularity). (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
CHOICE

Theoremdfac2 7967* Axiom of Choice (first form) of [Enderton] p. 49 implies of our Axiom of Choice (in the form of ac3 8298). The proof does not make use of AC. Note that the Axiom of Regularity is used by the proof. Specifically, elirrv 7521 and preleq 7528 that are referenced in the proof each make use of Regularity for their derivations. (The reverse implication can be derived without using Regularity; see dfac2a 7966.) TODO: Fix label in comment, and put label changes into list at top of set.mm. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
CHOICE

Theoremdfac7 7968* Equivalence of the Axiom of Choice (first form) of [Enderton] p. 49 and our Axiom of Choice (in the form of ac2 8297). The proof does not depend AC on but does depend on the Axiom of Regularity. (Contributed by Mario Carneiro, 17-May-2015.)
CHOICE

Theoremdfac0 7969* 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 8295. (Contributed by Mario Carneiro, 17-May-2015.)
CHOICE

Theoremdfac1 7970* Equivalence of two versions of the Axiom of Choice ax-ac 8295. 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.)
CHOICE

Theoremdfac8 7971* A proof of the equivalency of the Well Ordering Theorem weth 8331 and the Axiom of Choice ac7 8309. (Contributed by Mario Carneiro, 5-Jan-2013.)
CHOICE

Theoremdfac9 7972* Equivalence of the axiom of choice with a statement related to ac9 8319; definition AC3 of [Schechter] p. 139. (Contributed by Stefan O'Rear, 22-Feb-2015.)
CHOICE

Theoremdfac10 7973 Axiom of Choice equivalent: the cardinality function measures every set. (Contributed by Mario Carneiro, 6-May-2015.)
CHOICE

Theoremdfac10c 7974* Axiom of Choice equivalent: every set is equinumerous to an ordinal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
CHOICE

Theoremdfac10b 7975 Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 7953). (Contributed by Stefan O'Rear, 17-Jan-2015.)
CHOICE

Theoremacacni 7976 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
CHOICE AC

Theoremdfacacn 7977 A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
CHOICE AC

Theoremdfac13 7978 The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.)
CHOICE AC

Theoremdfac12lem1 7979* Lemma for dfac12 7985. (Contributed by Mario Carneiro, 29-May-2015.)
har       recs OrdIso               OrdIso

Theoremdfac12lem2 7980* Lemma for dfac12 7985. (Contributed by Mario Carneiro, 29-May-2015.)
har       recs OrdIso               OrdIso

Theoremdfac12lem3 7981* Lemma for dfac12 7985. (Contributed by Mario Carneiro, 29-May-2015.)
har       recs OrdIso

Theoremdfac12r 7982 The axiom of choice holds iff every ordinal has a well-orderable powerset. This version of dfac12 7985 does not assume the Axiom of Regularity. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremdfac12k 7983* Equivalence of dfac12 7985 and dfac12a 7984, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.)

Theoremdfac12a 7984 The axiom of choice holds iff every ordinal has a well-orderable powerset. (Contributed by Mario Carneiro, 29-May-2015.)
CHOICE

Theoremdfac12 7985 The axiom of choice holds iff every aleph has a well-orderable powerset. (Contributed by Mario Carneiro, 21-May-2015.)
CHOICE

Theoremkmlem1 7986* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, 1 => 2. (Contributed by NM, 5-Apr-2004.)

Theoremkmlem2 7987* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Theoremkmlem3 7988* 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.)

Theoremkmlem4 7989* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)

Theoremkmlem5 7990* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Theoremkmlem6 7991* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)

Theoremkmlem7 7992* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)

Theoremkmlem8 7993* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 4-Apr-2004.)

Theoremkmlem9 7994* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Theoremkmlem10 7995* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Theoremkmlem11 7996* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)

Theoremkmlem12 7997* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 27-Mar-2004.)

Theoremkmlem13 7998* Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 5-Apr-2004.)

Theoremkmlem14 7999* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)

Theoremkmlem15 8000* Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)

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