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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | acncc 8901 |
An ax-cc 8896 equivalent: every set has choice sets of
length ![]() |
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Theorem | axcc4dom 8902* | Relax the constraint on axcc4 8900 to dominance instead of equinumerosity. (Contributed by Mario Carneiro, 18-Jan-2014.) |
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Theorem | domtriomlem 8903* | Lemma for domtriom 8904. (Contributed by Mario Carneiro, 9-Feb-2013.) |
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Theorem | domtriom 8904 |
Trichotomy of equinumerosity for ![]() |
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Theorem | fin41 8905 | Under countable choice, the IV-finite sets (Dedekind-finite) coincide with I-finite (finite in the usual sense) sets. (Contributed by Mario Carneiro, 16-May-2015.) |
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Theorem | dominf 8906 |
A nonempty set that is a subset of its union is infinite. This version
is proved from ax-cc 8896. See dominfac 9029 for a version proved from
ax-ac 8920. The axiom of Regularity is used for this
proof, via
inf3lem6 8169, and its use is necessary: otherwise the set
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Axiom | ax-dc 8907* | Dependent Choice. Axiom DC1 of [Schechter] p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. Dependent choice is equivalent to the statement that every (nonempty) pruned tree has a branch. This axiom is redundant in ZFC; see axdc 8982. But ZF+DC is strictly weaker than ZF+AC, so this axiom provides for theorems that do not need the full power of AC. (Contributed by Mario Carneiro, 25-Jan-2013.) |
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Theorem | dcomex 8908 | The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.) |
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Theorem | axdc2lem 8909* |
Lemma for axdc2 8910. We construct a relation ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | axdc2 8910* |
An apparent strengthening of ax-dc 8907 (but derived from it) which shows
that there is a denumerable sequence ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | axdc3lem 8911* |
The class ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | axdc3lem2 8912* |
Lemma for axdc3 8915. We have constructed a "candidate
set" ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | axdc3lem3 8913* | Simple substitution lemma for axdc3 8915. (Contributed by Mario Carneiro, 27-Jan-2013.) |
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Theorem | axdc3lem4 8914* |
Lemma for axdc3 8915. We have constructed a "candidate
set" ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | axdc3 8915* |
Dependent Choice. Axiom DC1 of [Schechter]
p. 149, with the addition of
an initial value ![]() |
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Theorem | axdc4lem 8916* | Lemma for axdc4 8917. (Contributed by Mario Carneiro, 31-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2013.) |
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Theorem | axdc4 8917* |
A more general version of axdc3 8915 that allows the function ![]() ![]() |
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Theorem | axcclem 8918* | Lemma for axcc 8919. (Contributed by Mario Carneiro, 2-Feb-2013.) (Revised by Mario Carneiro, 16-Nov-2013.) |
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Theorem | axcc 8919* | Although CC can be proven trivially using ac5 8938, we prove it here using DC. (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Feb-2013.) |
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Axiom | ax-ac 8920* |
Axiom of Choice. The Axiom of Choice (AC) is usually considered an
extension of ZF set theory rather than a proper part of it. It is
sometimes considered philosophically controversial because it asserts
the existence of a set without telling us what the set is. ZF set
theory that includes AC is called ZFC.
The unpublished version given here says that given any set This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 8926 is slightly shorter when the biconditional of ax-ac 8920 is expanded into implication and negation. In axac3 8925 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9137 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 8953, ac5 8938, and ac7 8934. The Axiom of Regularity ax-reg 8138 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 8592. Equivalents to AC are the well-ordering theorem weth 8956 and Zorn's lemma zorn 8968. See ac4 8936 for comments about stronger versions of AC. In order to avoid uses of ax-reg 8138 for derivation of AC equivalents, we provide ax-ac2 8924 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8924 from ax-ac 8920 is shown by theorem axac2 8927, and the reverse derivation by axac 8928. Therefore, new proofs should normally use ax-ac2 8924 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.) |
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Theorem | zfac 8921* | Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 8920. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.) |
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Theorem | ac2 8922* | Axiom of Choice equivalent. By using restricted quantifiers, we can express the Axiom of Choice with a single explicit conjunction. (If you want to figure it out, the rewritten equivalent ac3 8923 is easier to understand.) Note: aceq0 8580 shows the logical equivalence to ax-ac 8920. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.) |
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Theorem | ac3 8923* |
Axiom of Choice using abbreviations. The logical equivalence to ax-ac 8920
can be established by chaining aceq0 8580 and aceq2 8581. A standard
textbook version of AC is derived from this one in dfac2a 8591, and this
version of AC is derived from the textbook version in dfac2 8592.
The following sketch will help you understand this version of the
axiom. Given any set
For example, suppose
(New usage is discouraged.) (Contributed by NM, 19-Jul-1996.) |
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Axiom | ax-ac2 8924* | In order to avoid uses of ax-reg 8138 for derivation of AC equivalents, we provide ax-ac2 8924, which is equivalent to the standard AC of textbooks. This appears to be the shortest known equivalent to the standard AC when expressed in terms of set theory primitives. It was found by Kurt Maes as theorem ackm 8926. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1680 available. The derivation of ax-ac2 8924 from ax-ac 8920 is shown by theorem axac2 8927, and the reverse derivation by axac 8928. Note that we use ax-reg 8138 to derive ax-ac 8920 from ax-ac2 8924, but not to derive ax-ac2 8924 from ax-ac 8920. (Contributed by NM, 19-Dec-2016.) |
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Theorem | axac3 8925 | This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8924 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.) |
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Theorem | ackm 8926* |
A remarkable equivalent to the Axiom of Choice that has only five
quantifiers (when expanded to ![]() ![]() ![]() The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.) |
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Theorem | axac2 8927* | Derive ax-ac2 8924 from ax-ac 8920. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | axac 8928* | Derive ax-ac 8920 from ax-ac2 8924. Note that ax-reg 8138 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.) |
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Theorem | axaci 8929 | Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.) |
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Theorem | cardeqv 8930 | All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.) |
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Theorem | numth3 8931 | All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
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Theorem | numth2 8932* | Numeration theorem: any set is equinumerous to some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 20-Oct-2003.) |
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Theorem | numth 8933* | Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Mario Carneiro, 8-Jan-2015.) |
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Theorem | ac7 8934* | An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.) |
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Theorem | ac7g 8935* | An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.) |
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Theorem | ac4 8936* |
Equivalent of Axiom of Choice. We do not insist that ![]() ![]()
Takeuti and Zaring call this "weak choice" in contrast to
"strong
choice" Weak choice can be strengthened in a different direction to choose from a collection of proper classes; see ac6s5 8952. (Contributed by NM, 21-Jul-1996.) |
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Theorem | ac4c 8937* | Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997.) |
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Theorem | ac5 8938* |
An Axiom of Choice equivalent: there exists a function ![]() ![]() ![]() |
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Theorem | ac5b 8939* | Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.) |
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Theorem | ac6num 8940* | A version of ac6 8941 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.) |
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Theorem | ac6 8941* |
Equivalent of Axiom of Choice. This is useful for proving that there
exists, for example, a sequence mapping natural numbers to members of a
larger set ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ac6c4 8942* |
Equivalent of Axiom of Choice. ![]() ![]() ![]() ![]() ![]() |
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Theorem | ac6c5 8943* |
Equivalent of Axiom of Choice. ![]() ![]() ![]() ![]() ![]() |
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Theorem | ac9 8944* | An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. (Contributed by Mario Carneiro, 22-Mar-2013.) |
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Theorem | ac6s 8945* |
Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 8395,
we
derive this strong version of ac6 8941 that doesn't require ![]() |
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Theorem | ac6n 8946* | Equivalent of Axiom of Choice. Contrapositive of ac6s 8945. (Contributed by NM, 10-Jun-2007.) |
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Theorem | ac6s2 8947* | Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 8948. (Contributed by NM, 29-Sep-2006.) |
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Theorem | ac6s3 8948* | Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.) |
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Theorem | ac6sg 8949* | ac6s 8945 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.) |
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Theorem | ac6sf 8950* | Version of ac6 8941 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) |
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Theorem | ac6s4 8951* |
Generalization of the Axiom of Choice to proper classes. ![]() ![]() ![]() ![]() ![]() |
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Theorem | ac6s5 8952* |
Generalization of the Axiom of Choice to proper classes. ![]() ![]() ![]() ![]() ![]() |
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Theorem | ac8 8953* |
An Axiom of Choice equivalent. Given a family ![]() ![]() |
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Theorem | ac9s 8954* |
An Axiom of Choice equivalent: the infinite Cartesian product of
nonempty classes is nonempty. Axiom of Choice (second form) of
[Enderton] p. 55 and its converse.
This is a stronger version of the
axiom in Enderton, with no existence requirement for the family of
classes ![]() ![]() ![]() ![]() |
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Theorem | numthcor 8955* | Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.) |
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Theorem | weth 8956* |
Well-ordering theorem: any set ![]() |
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Theorem | zorn2lem1 8957* | Lemma for zorn2 8967. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
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Theorem | zorn2lem2 8958* | Lemma for zorn2 8967. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
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Theorem | zorn2lem3 8959* | Lemma for zorn2 8967. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
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Theorem | zorn2lem4 8960* | Lemma for zorn2 8967. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
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Theorem | zorn2lem5 8961* | Lemma for zorn2 8967. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
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Theorem | zorn2lem6 8962* | Lemma for zorn2 8967. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
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Theorem | zorn2lem7 8963* | Lemma for zorn2 8967. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
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Theorem | zorn2g 8964* |
Zorn's Lemma of [Monk1] p. 117. This version of
zorn2 8967 avoids the
Axiom of Choice by assuming that ![]() |
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Theorem | zorng 8965* |
Zorn's Lemma. If the union of every chain (with respect to inclusion)
in a set belongs to the set, then the set contains a maximal element.
Theorem 6M of [Enderton] p. 151. This
version of zorn 8968 avoids the
Axiom of Choice by assuming that ![]() |
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Theorem | zornn0g 8966* |
Variant of Zorn's lemma zorng 8965 in which ![]() ![]() |
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Theorem | zorn2 8967* |
Zorn's Lemma of [Monk1] p. 117. This theorem is
equivalent to the Axiom
of Choice and states that every partially ordered set ![]() ![]() |
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Theorem | zorn 8968* | Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 8967 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.) |
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Theorem | zornn0 8969* |
Variant of Zorn's lemma zorn 8968 in which ![]() ![]() |
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Theorem | ttukeylem1 8970* | Lemma for ttukey 8979. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.) |
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Theorem | ttukeylem2 8971* | Lemma for ttukey 8979. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.) |
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Theorem | ttukeylem3 8972* | Lemma for ttukey 8979. (Contributed by Mario Carneiro, 11-May-2015.) |
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Theorem | ttukeylem4 8973* | Lemma for ttukey 8979. (Contributed by Mario Carneiro, 15-May-2015.) |
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Theorem | ttukeylem5 8974* |
Lemma for ttukey 8979. The ![]() |
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Theorem | ttukeylem6 8975* | Lemma for ttukey 8979. (Contributed by Mario Carneiro, 15-May-2015.) |
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Theorem | ttukeylem7 8976* | Lemma for ttukey 8979. (Contributed by Mario Carneiro, 15-May-2015.) |
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Theorem | ttukey2g 8977* |
The Teichmüller-Tukey Lemma ttukey 8979 with a slightly stronger
conclusion: we can set up the maximal element of ![]() ![]() ![]() ![]() |
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Theorem | ttukeyg 8978* |
The Teichmüller-Tukey Lemma ttukey 8979 stated with the "choice" as
an
antecedent (the hypothesis ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ttukey 8979* |
The Teichmüller-Tukey Lemma, an Axiom of Choice equivalent. If
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Theorem | axdclem 8980* | Lemma for axdc 8982. (Contributed by Mario Carneiro, 25-Jan-2013.) |
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Theorem | axdclem2 8981* |
Lemma for axdc 8982. Using the full Axiom of Choice, we can
construct a
choice function ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | axdc 8982* | This theorem derives ax-dc 8907 using ax-ac 8920 and ax-inf 8174. 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.) |
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Theorem | fodom 8983 | An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8935. AC is not needed for finite sets - see fodomfi 7881. See also fodomnum 8519. (Contributed by NM, 23-Jul-2004.) |
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Theorem | fodomg 8984 | An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.) |
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Theorem | fodomb 8985* | Equivalence of an onto mapping and dominance for a nonempty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.) |
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Theorem | wdomac 8986 | 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.) |
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Theorem | brdom3 8987* | Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.) |
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Theorem | brdom5 8988* | An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.) |
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Theorem | brdom4 8989* | An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
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Theorem | brdom7disj 8990* | An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
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Theorem | brdom6disj 8991* | An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.) |
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Theorem | fin71ac 8992 | 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.) |
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Theorem | imadomg 8993 | 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.) |
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Theorem | fnrndomg 8994 | The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.) |
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Theorem | iunfo 8995* | 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.) |
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Theorem | iundom2g 8996* |
An upper bound for the cardinality of a disjoint indexed union, with
explicit choice principles. ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | iundomg 8997* |
An upper bound for the cardinality of an indexed union, with explicit
choice principles. ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | iundom 8998* |
An upper bound for the cardinality of an indexed union. ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | unidom 8999* | 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.) |
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Theorem | uniimadom 9000* | 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.) |
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