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Type | Label | Description |
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Statement | ||
Theorem | difinfsn 7101* | An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.) |
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Theorem | difinfinf 7102* | An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.) |
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Syntax | cdjud 7103 | Syntax for the domain-disjoint-union of two relations. |
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Definition | df-djud 7104 |
The "domain-disjoint-union" of two relations: if ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
Remark: the restrictions to |
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Theorem | djufun 7105 | The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.) |
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Theorem | djudm 7106 | The domain of the "domain-disjoint-union" is the disjoint union of the domains. Remark: its range is the (standard) union of the ranges. (Contributed by BJ, 10-Jul-2022.) |
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Theorem | djuinj 7107 | The "domain-disjoint-union" of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.) |
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Theorem | 0ct 7108 | The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.) |
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Theorem | ctmlemr 7109* | Lemma for ctm 7110. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.) |
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Theorem | ctm 7110* | 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 7111* |
Lemma for ctssdc 7114. The ![]() ![]() ![]() ![]() |
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Theorem | ctssdccl 7112* |
A mapping from a decidable subset of the natural numbers onto a
countable set. This is similar to one direction of ctssdc 7114 but
expressed in terms of classes rather than ![]() |
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Theorem | ctssdclemr 7113* | Lemma for ctssdc 7114. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.) |
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Theorem | ctssdc 7114* | 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 7150. (Contributed by Jim Kingdon, 15-Aug-2023.) |
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Theorem | enumctlemm 7115* |
Lemma for enumct 7116. The case where ![]() |
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Theorem | enumct 7116* |
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 7117* | A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.) |
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Theorem | omct 7118 |
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Theorem | ctfoex 7119* | A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.) |
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This section introduces the one-point compactification of the set of natural
numbers, introduced by Escardo as the set of nonincreasing sequences on
| ||
Syntax | xnninf 7120 |
Set of nonincreasing sequences in ![]() ![]() ![]() |
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Definition | df-nninf 7121* |
Define the set of nonincreasing sequences in ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nninfex 7122 | ℕ∞ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.) |
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Theorem | nninff 7123 | An element of ℕ∞ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.) |
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Theorem | infnninf 7124 |
The point at infinity in ℕ∞ is the constant sequence
equal to
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | infnninfOLD 7125 | Obsolete version of infnninf 7124 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | nnnninf 7126* |
Elements of ℕ∞ corresponding to natural numbers. The
natural
number ![]() ![]() |
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Theorem | nnnninf2 7127* |
Canonical embedding of ![]() ![]() |
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Theorem | nnnninfeq 7128* | Mapping of a natural number to an element of ℕ∞. (Contributed by Jim Kingdon, 4-Aug-2022.) |
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Theorem | nnnninfeq2 7129* |
Mapping of a natural number to an element of ℕ∞.
Similar to
nnnninfeq 7128 but if we have information about a single
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Theorem | nninfisollem0 7130* |
Lemma for nninfisol 7133. The case where ![]() |
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Theorem | nninfisollemne 7131* |
Lemma for nninfisol 7133. A case where ![]() ![]() ![]() |
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Theorem | nninfisollemeq 7132* |
Lemma for nninfisol 7133. The case where ![]() ![]() ![]() |
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Theorem | nninfisol 7133* |
Finite elements of ℕ∞ are isolated. That is, given a
natural
number and any element of ℕ∞, it is decidable
whether the
natural number (when converted to an element of
ℕ∞) is equal to
the given element of ℕ∞. Stated in an online
post by Martin
Escardo. One way to understand this theorem is that you do not need to
look at an unbounded number of elements of the sequence ![]() ![]() ![]() |
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Syntax | comni 7134 | Extend class definition to include the class of omniscient sets. |
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Definition | df-omni 7135* |
An omniscient set is one where we can decide whether a predicate (here
represented by a function ![]() ![]() ![]()
In particular, |
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Theorem | isomni 7136* | The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.) |
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Theorem | isomnimap 7137* | The predicate of being omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 13-Jul-2022.) |
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Theorem | enomnilem 7138 | Lemma for enomni 7139. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.) |
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Theorem | enomni 7139 |
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 7140 | 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 7141 | Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7142. (Contributed by Jim Kingdon, 29-Jun-2022.) |
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Theorem | exmidomni 7142 | Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.) |
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Theorem | exmidlpo 7143 | Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.) |
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Theorem | fodjuomnilemdc 7144* | Lemma for fodjuomni 7149. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.) |
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Theorem | fodjuf 7145* |
Lemma for fodjuomni 7149 and fodjumkv 7160. Domain and range of ![]() |
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Theorem | fodjum 7146* |
Lemma for fodjuomni 7149 and fodjumkv 7160. A condition which shows that
![]() |
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Theorem | fodju0 7147* |
Lemma for fodjuomni 7149 and fodjumkv 7160. A condition which shows that
![]() |
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Theorem | fodjuomnilemres 7148* |
Lemma for fodjuomni 7149. The final result with ![]() |
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Theorem | fodjuomni 7149* |
A condition which ensures ![]() |
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Theorem | ctssexmid 7150* | The decidability condition in ctssdc 7114 is needed. More specifically, ctssdc 7114 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 7151 | Extend class definition to include the class of Markov sets. |
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Definition | df-markov 7152* |
A Markov set is one where if a predicate (here represented by a function
![]() ![]() ![]()
In particular, |
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Theorem | ismkv 7153* | The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.) |
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Theorem | ismkvmap 7154* | The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.) |
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Theorem | ismkvnex 7155* |
The predicate of being Markov stated in terms of double negation and
comparison with ![]() |
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Theorem | omnimkv 7156 |
An omniscient set is Markov. In particular, the case where ![]() ![]() |
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Theorem | exmidmp 7157 | Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.) |
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Theorem | mkvprop 7158* |
Markov's Principle expressed in terms of propositions (or more
precisely, the ![]() ![]() ![]() |
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Theorem | fodjumkvlemres 7159* |
Lemma for fodjumkv 7160. The final result with ![]() |
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Theorem | fodjumkv 7160* | A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.) |
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Theorem | enmkvlem 7161 | Lemma for enmkv 7162. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.) |
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Theorem | enmkv 7162 |
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 7163 | Extend class definition to include the class of weakly omniscient sets. |
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Definition | df-womni 7164* |
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 7165* | The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.) |
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Theorem | iswomnimap 7166* | 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 7167 |
A set is omniscient if and only if it is weakly omniscient and Markov.
The case ![]() ![]() ![]() ![]() ![]() |
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Theorem | lpowlpo 7168 | LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7167. There is an analogue in terms of analytic omniscience principles at tridceq 14889. (Contributed by Jim Kingdon, 24-Jul-2024.) |
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Theorem | enwomnilem 7169 | Lemma for enwomni 7170. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.) |
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Theorem | enwomni 7170 |
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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Theorem | nninfdcinf 7171* | The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ∞ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.) |
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Theorem | nninfwlporlemd 7172* | Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.) |
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Theorem | nninfwlporlem 7173* | Lemma for nninfwlpor 7174. The result. (Contributed by Jim Kingdon, 7-Dec-2024.) |
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Theorem | nninfwlpor 7174* | The Weak Limited Principle of Omniscience (WLPO) implies that equality for ℕ∞ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.) |
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Theorem | nninfwlpoimlemg 7175* | Lemma for nninfwlpoim 7178. (Contributed by Jim Kingdon, 8-Dec-2024.) |
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Theorem | nninfwlpoimlemginf 7176* | Lemma for nninfwlpoim 7178. (Contributed by Jim Kingdon, 8-Dec-2024.) |
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Theorem | nninfwlpoimlemdc 7177* | Lemma for nninfwlpoim 7178. (Contributed by Jim Kingdon, 8-Dec-2024.) |
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Theorem | nninfwlpoim 7178* | Decidable equality for ℕ∞ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.) |
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Theorem | nninfwlpo 7179* | Decidability of equality for ℕ∞ is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.) |
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Syntax | ccrd 7180 | Extend class definition to include the cardinal size function. |
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Definition | df-card 7181* | 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 7182* | The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.) |
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Theorem | isnumi 7183 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
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Theorem | finnum 7184 | 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 7185 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
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Theorem | cardval3ex 7186* |
The value of ![]() ![]() ![]() ![]() ![]() |
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Theorem | oncardval 7187* | 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 7188 | 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 7189 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
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Theorem | carden2bex 7190* | 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 7191 | Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.) |
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Theorem | pr2nelem 7192 | Lemma for pr2ne 7193. (Contributed by FL, 17-Aug-2008.) |
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Theorem | pr2ne 7193 | If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.) |
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Theorem | exmidonfinlem 7194* | Lemma for exmidonfin 7195. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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Theorem | exmidonfin 7195 | 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 6874 and nnon 4611. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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Theorem | en2eleq 7196 | 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 7197 | 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 7198 | Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.) |
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Theorem | infpwfidom 7199 |
The collection of finite subsets of a set dominates the set. (We use
the weaker sethood assumption ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | exmidfodomrlemeldju 7200 | Lemma for exmidfodomr 7205. A variant of djur 7070. (Contributed by Jim Kingdon, 2-Jul-2022.) |
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