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Type | Label | Description |
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Statement | ||
Theorem | ctmlemr 7001* | Lemma for ctm 7002. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.) |
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Theorem | ctm 7002* | Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.) |
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Theorem | ctssdclemn0 7003* |
Lemma for ctssdc 7006. The ![]() ![]() ![]() ![]() |
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Theorem | ctssdccl 7004* |
A mapping from a decidable subset of the natural numbers onto a
countable set. This is similar to one direction of ctssdc 7006 but
expressed in terms of classes rather than ![]() |
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Theorem | ctssdclemr 7005* | Lemma for ctssdc 7006. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.) |
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Theorem | ctssdc 7006* | A set is countable iff there is a surjection from a decidable subset of the natural numbers onto it. The decidability condition is needed as shown at ctssexmid 7032. (Contributed by Jim Kingdon, 15-Aug-2023.) |
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Theorem | enumctlemm 7007* |
Lemma for enumct 7008. The case where ![]() |
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Theorem | enumct 7008* |
A finitely enumerable set is countable. Lemma 8.1.14 of [AczelRathjen],
p. 73 (except that our definition of countable does not require the set
to be inhabited). "Finitely enumerable" is defined as
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Theorem | finct 7009* | A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.) |
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Theorem | omct 7010 |
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Theorem | ctfoex 7011* | A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.) |
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Syntax | comni 7012 | Extend class definition to include the class of omniscient sets. |
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Syntax | xnninf 7013 |
Set of nonincreasing sequences in ![]() ![]() ![]() |
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Definition | df-omni 7014* |
An omniscient set is one where we can decide whether a predicate (here
represented by a function ![]() ![]() ![]()
In particular, |
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Definition | df-nninf 7015* |
Define the set of nonincreasing sequences in ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | isomni 7016* | The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.) |
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Theorem | isomnimap 7017* | The predicate of being omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 13-Jul-2022.) |
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Theorem | enomnilem 7018 | Lemma for enomni 7019. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.) |
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Theorem | enomni 7019 |
Omniscience is invariant with respect to equinumerosity. For example,
this means that we can express the Limited Principle of Omniscience as
either ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | finomni 7020 | A finite set is omniscient. Remark right after Definition 3.1 of [Pierik], p. 14. (Contributed by Jim Kingdon, 28-Jun-2022.) |
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Theorem | exmidomniim 7021 | Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7022. (Contributed by Jim Kingdon, 29-Jun-2022.) |
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Theorem | exmidomni 7022 | Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.) |
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Theorem | exmidlpo 7023 | Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.) |
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Theorem | fodjuomnilemdc 7024* | Lemma for fodjuomni 7029. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.) |
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Theorem | fodjuf 7025* |
Lemma for fodjuomni 7029 and fodjumkv 7042. Domain and range of ![]() |
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Theorem | fodjum 7026* |
Lemma for fodjuomni 7029 and fodjumkv 7042. A condition which shows that
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Theorem | fodju0 7027* |
Lemma for fodjuomni 7029 and fodjumkv 7042. A condition which shows that
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Theorem | fodjuomnilemres 7028* |
Lemma for fodjuomni 7029. The final result with ![]() |
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Theorem | fodjuomni 7029* |
A condition which ensures ![]() |
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Theorem | infnninf 7030 |
The point at infinity in ℕ∞ (the constant sequence
equal to
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Theorem | nnnninf 7031* |
Elements of ℕ∞ corresponding to natural numbers. The
natural
number ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ctssexmid 7032* | The decidability condition in ctssdc 7006 is needed. More specifically, ctssdc 7006 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.) |
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Syntax | cmarkov 7033 | Extend class definition to include the class of Markov sets. |
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Definition | df-markov 7034* |
A Markov set is one where if a predicate (here represented by a function
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In particular, |
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Theorem | ismkv 7035* | The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.) |
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Theorem | ismkvmap 7036* | The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.) |
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Theorem | ismkvnex 7037* |
The predicate of being Markov stated in terms of double negation and
comparison with ![]() |
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Theorem | omnimkv 7038 |
An omniscient set is Markov. In particular, the case where ![]() ![]() |
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Theorem | exmidmp 7039 | Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.) |
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Theorem | mkvprop 7040* |
Markov's Principle expressed in terms of propositions (or more
precisely, the ![]() ![]() ![]() |
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Theorem | fodjumkvlemres 7041* |
Lemma for fodjumkv 7042. The final result with ![]() |
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Theorem | fodjumkv 7042* | A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.) |
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Theorem | enmkvlem 7043 | Lemma for enmkv 7044. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.) |
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Theorem | enmkv 7044 |
Being Markov is invariant with respect to equinumerosity. For example,
this means that we can express the Markov's Principle as either
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Syntax | cwomni 7045 | Extend class definition to include the class of weakly omniscient sets. |
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Definition | df-womni 7046* |
A weakly omniscient set is one where we can decide whether a predicate
(here represented by a function ![]() ![]()
In particular, The term WLPO is common in the literature; there appears to be no widespread term for what we are calling a weakly omniscient set. (Contributed by Jim Kingdon, 9-Jun-2024.) |
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Theorem | iswomni 7047* | The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.) |
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Theorem | iswomnimap 7048* | The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.) |
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Theorem | omniwomnimkv 7049 |
A set is omniscient if and only if it is weakly omniscient and Markov.
The case ![]() ![]() ![]() ![]() ![]() |
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Theorem | enwomnilem 7050 | Lemma for enwomni 7051. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.) |
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Theorem | enwomni 7051 |
Weak omniscience is invariant with respect to equinumerosity. For
example, this means that we can express the Weak Limited Principle of
Omniscience as either ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Syntax | ccrd 7052 | Extend class definition to include the cardinal size function. |
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Definition | df-card 7053* | Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.) |
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Theorem | cardcl 7054* | The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.) |
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Theorem | isnumi 7055 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
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Theorem | finnum 7056 | Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
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Theorem | onenon 7057 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
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Theorem | cardval3ex 7058* |
The value of ![]() ![]() ![]() ![]() ![]() |
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Theorem | oncardval 7059* | The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
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Theorem | cardonle 7060 | The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) |
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Theorem | card0 7061 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
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Theorem | carden2bex 7062* | If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.) |
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Theorem | pm54.43 7063 | Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.) |
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Theorem | pr2nelem 7064 | Lemma for pr2ne 7065. (Contributed by FL, 17-Aug-2008.) |
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Theorem | pr2ne 7065 | If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.) |
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Theorem | exmidonfinlem 7066* | Lemma for exmidonfin 7067. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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Theorem | exmidonfin 7067 | If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6774 and nnon 4531. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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Theorem | en2eleq 7068 | Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
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Theorem | en2other2 7069 | Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
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Theorem | dju1p1e2 7070 | Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.) |
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Theorem | infpwfidom 7071 |
The collection of finite subsets of a set dominates the set. (We use
the weaker sethood assumption ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | exmidfodomrlemeldju 7072 | Lemma for exmidfodomr 7077. A variant of djur 6962. (Contributed by Jim Kingdon, 2-Jul-2022.) |
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Theorem | exmidfodomrlemreseldju 7073 | Lemma for exmidfodomrlemrALT 7076. A variant of eldju 6961. (Contributed by Jim Kingdon, 9-Jul-2022.) |
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Theorem | exmidfodomrlemim 7074* | Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.) |
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Theorem | exmidfodomrlemr 7075* | The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.) |
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Theorem | exmidfodomrlemrALT 7076* | The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. An alternative proof of exmidfodomrlemr 7075. In particular, this proof uses eldju 6961 instead of djur 6962 and avoids djulclb 6948. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.) |
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Theorem | exmidfodomr 7077* | Excluded middle is equivalent to the existence of a mapping from any set onto any inhabited set that it dominates. (Contributed by Jim Kingdon, 1-Jul-2022.) |
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Syntax | wac 7078 | Formula for an abbreviation of the axiom of choice. |
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Definition | df-ac 7079* |
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 are some decisions about how to write this definition especially around whether ax-setind 4460 is needed to show equivalence to other ways of stating choice, and about whether choice functions are available for nonempty sets or inhabited sets. (Contributed by Mario Carneiro, 22-Feb-2015.) |
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Theorem | acfun 7080* | A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.) |
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Theorem | exmidaclem 7081* | Lemma for exmidac 7082. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.) |
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Theorem | exmidac 7082 | The axiom of choice implies excluded middle. See acexmid 5781 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.) |
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Theorem | endjudisj 7083 | Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.) |
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Theorem | djuen 7084 | Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.) |
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Theorem | djuenun 7085 | Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.) |
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Theorem | dju1en 7086 | 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.) |
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Theorem | dju0en 7087 | 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.) |
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Theorem | xp2dju 7088 | 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.) |
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Theorem | djucomen 7089 | 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.) |
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Theorem | djuassen 7090 | 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.) |
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Theorem | xpdjuen 7091 | 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.) |
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Theorem | djudoml 7092 | A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.) |
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Theorem | djudomr 7093 | A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.) |
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We have already introduced the full Axiom of Choice df-ac 7079 but since it implies excluded middle as shown at exmidac 7082, it is not especially relevant to us. In this section we define countable choice and dependent choice, which are not as strong as thus often considered in mathematics which seeks to avoid full excluded middle. | ||
Syntax | wacc 7094 | Formula for an abbreviation of countable choice. |
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Definition | df-cc 7095* | The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7079 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.) |
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Theorem | ccfunen 7096* | Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.) |
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Theorem | cc1 7097* | Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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Theorem | cc2lem 7098* | Lemma for cc2 7099. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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Theorem | cc2 7099* | Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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Theorem | cc3 7100* | Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.) |
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