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Type | Label | Description |
---|---|---|
Statement | ||
In this section, we state the axiom scheme of subset collection, which is part of CZF set theory. | ||
Axiom | ax-sscoll 13401* | Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that represents a multivalued function from to , or equivalently a collection of nonempty subsets of indexed by , and the consequent asserts the existence of a subset of which "collects" at least one element in the image of each and which is made only of such elements. The axiom asserts the existence, for any sets , of a set such that that implication holds for any value of the parameter of . (Contributed by BJ, 5-Oct-2019.) |
Theorem | sscoll2 13402* | Version of ax-sscoll 13401 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.) |
Axiom | ax-ddkcomp 13403 | Axiom of Dedekind completeness for Dedekind real numbers: every inhabited upper-bounded located set of reals has a real upper bound. Ideally, this axiom should be "proved" as "axddkcomp" for the real numbers constructed from IZF, and then the axiom ax-ddkcomp 13403 should be used in place of construction specific results. In particular, axcaucvg 7761 should be proved from it. (Contributed by BJ, 24-Oct-2021.) |
Theorem | el2oss1o 13404 | Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 13405. (Contributed by Jim Kingdon, 8-Aug-2022.) |
Theorem | ss1oel2o 13405 | Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4132 which more directly illustrates the contrast with el2oss1o 13404. (Contributed by Jim Kingdon, 8-Aug-2022.) |
EXMID | ||
Theorem | nnti 13406 | Ordering on a natural number generates a tight apartness. (Contributed by Jim Kingdon, 7-Aug-2022.) |
Theorem | 012of 13407 | Mapping zero and one between and style integers. (Contributed by Jim Kingdon, 28-Jun-2024.) |
frec | ||
Theorem | 2o01f 13408 | Mapping zero and one between and style integers. (Contributed by Jim Kingdon, 28-Jun-2024.) |
frec | ||
Theorem | pwtrufal 13409 | A subset of the singleton cannot be anything other than or . Removing the double negation would change the meaning, as seen at exmid01 4132. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4130), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.) |
Theorem | pwle2 13410* | An exercise related to copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.) |
Theorem | pwf1oexmid 13411* | An exercise related to copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.) |
EXMID | ||
Theorem | exmid1stab 13412* | If any proposition is stable, excluded middle follows. We are thinking of as a proposition and as "x is true". (Contributed by Jim Kingdon, 28-Nov-2023.) |
STAB EXMID | ||
Theorem | subctctexmid 13413* | If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.) |
⊔ Markov EXMID | ||
Theorem | sssneq 13414* | Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.) |
Theorem | pw1nct 13415* | A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.) |
⊔ | ||
Theorem | 0nninf 13416 | The zero element of ℕ_{∞} (the constant sequence equal to ). (Contributed by Jim Kingdon, 14-Jul-2022.) |
ℕ_{∞} | ||
Theorem | nninff 13417 | An element of ℕ_{∞} is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nnsf 13418* | Domain and range of . Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.) |
ℕ_{∞} ℕ_{∞}ℕ_{∞} | ||
Theorem | peano4nninf 13419* | The successor function on ℕ_{∞} is one to one. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 31-Jul-2022.) |
ℕ_{∞} ℕ_{∞}ℕ_{∞} | ||
Theorem | peano3nninf 13420* | The successor function on ℕ_{∞} is never zero. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfalllemn 13421* | Lemma for nninfall 13423. Mapping of a natural number to an element of ℕ_{∞}. (Contributed by Jim Kingdon, 4-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nninfalllem1 13422* | Lemma for nninfall 13423. (Contributed by Jim Kingdon, 1-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfall 13423* | Given a decidable predicate on ℕ_{∞}, showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which is a decidable predicate is that it assigns a value of either or (which can be thought of as false and true) to every element of ℕ_{∞}. Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfex 13424 | ℕ_{∞} is a set. (Contributed by Jim Kingdon, 10-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nninfsellemdc 13425* | Lemma for nninfself 13428. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.) |
ℕ_{∞} DECID | ||
Theorem | nninfsellemcl 13426* | Lemma for nninfself 13428. (Contributed by Jim Kingdon, 8-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nninfsellemsuc 13427* | Lemma for nninfself 13428. (Contributed by Jim Kingdon, 6-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nninfself 13428* | Domain and range of the selection function for ℕ_{∞}. (Contributed by Jim Kingdon, 6-Aug-2022.) |
ℕ_{∞} ℕ_{∞}ℕ_{∞} | ||
Theorem | nninfsellemeq 13429* | Lemma for nninfsel 13432. (Contributed by Jim Kingdon, 9-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfsellemqall 13430* | Lemma for nninfsel 13432. (Contributed by Jim Kingdon, 9-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfsellemeqinf 13431* | Lemma for nninfsel 13432. (Contributed by Jim Kingdon, 9-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfsel 13432* | is a selection function for ℕ_{∞}. Theorem 3.6 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 9-Aug-2022.) |
ℕ_{∞} ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfomnilem 13433* | Lemma for nninfomni 13434. (Contributed by Jim Kingdon, 10-Aug-2022.) |
ℕ_{∞} ℕ_{∞} Omni | ||
Theorem | nninfomni 13434 | ℕ_{∞} is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.) |
ℕ_{∞} Omni | ||
Theorem | nninffeq 13435* | Equality of two functions on ℕ_{∞} which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one, . (Contributed by Jim Kingdon, 4-Aug-2023.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | exmidsbthrlem 13436* | Lemma for exmidsbthr 13437. (Contributed by Jim Kingdon, 11-Aug-2022.) |
ℕ_{∞} EXMID | ||
Theorem | exmidsbthr 13437* | The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.) |
EXMID | ||
Theorem | exmidsbth 13438* |
The Schroeder-Bernstein Theorem is equivalent to excluded middle. This
is Metamath 100 proof #25. The forward direction (isbth 6872) is the
proof of the Schroeder-Bernstein Theorem from the Metamath Proof
Explorer database (in which excluded middle holds), but adapted to use
EXMID as an antecedent rather than being unconditionally
true, as in
the non-intuitionist proof at
https://us.metamath.org/mpeuni/sbth.html 6872.
The reverse direction (exmidsbthr 13437) is the one which establishes that Schroeder-Bernstein implies excluded middle. This resolves the question of whether we will be able to prove Schroeder-Bernstein from our axioms in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.) |
EXMID | ||
Theorem | sbthomlem 13439 | Lemma for sbthom 13440. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.) |
Omni ⊔ | ||
Theorem | sbthom 13440 | Schroeder-Bernstein is not possible even for . We know by exmidsbth 13438 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is ? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.) |
Omni EXMID | ||
Theorem | qdencn 13441* | The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 11035 (and also would hold for with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | refeq 13442* | Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.) |
Theorem | triap 13443 | Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.) |
DECID # | ||
Theorem | isomninnlem 13444* | Lemma for isomninn 13445. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.) |
frec Omni | ||
Theorem | isomninn 13445* | Omniscience stated in terms of natural numbers. Similar to isomnimap 7030 but it will sometimes be more convenient to use and rather than and . (Contributed by Jim Kingdon, 30-Aug-2023.) |
Omni | ||
Theorem | cvgcmp2nlemabs 13446* | Lemma for cvgcmp2n 13447. The partial sums get closer to each other as we go further out. The proof proceeds by rewriting as the sum of and a term which gets smaller as gets large. (Contributed by Jim Kingdon, 25-Aug-2023.) |
Theorem | cvgcmp2n 13447* | A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.) |
Theorem | iooref1o 13448 | A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.) |
Theorem | iooreen 13449 | An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.) |
Omniscience principles refer to several propositions, most of them weaker than full excluded middle, which do not follow from the axioms of IZF set theory. They are: (0) the Principle of Omniscience (PO), which is another name for excluded middle (see exmidomni 7035), (1) the Limited Principle of Omniscience (LPO) is Omni (see df-omni 7028), (2) the Weak Limited Principle of Omniscience (WLPO) is WOmni (see df-womni 7057), (3) Markov's Principle (MP) is Markov (see df-markov 7045), (4) the Lesser Limited Principle of Omniscience (LLPO) is not yet defined in iset.mm. They also have analytic counterparts each of which follows from the corresponding omniscience principle: (1) Analytic LPO is real number trichotomy, (see trilpo 13457), (2) Analytic WLPO is decidability of real number equality, DECID (see redcwlpo 13469), (3) Analytic MP is # (see neapmkv 13481), (4) Analytic LLPO is real number dichotomy, (most relevant current theorem is maxclpr 11055). | ||
Theorem | trilpolemclim 13450* | Lemma for trilpo 13457. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.) |
Theorem | trilpolemcl 13451* | Lemma for trilpo 13457. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.) |
Theorem | trilpolemisumle 13452* | Lemma for trilpo 13457. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.) |
Theorem | trilpolemgt1 13453* | Lemma for trilpo 13457. The case. (Contributed by Jim Kingdon, 23-Aug-2023.) |
Theorem | trilpolemeq1 13454* | Lemma for trilpo 13457. The case. This is proved by noting that if any is zero, then the infinite sum is less than one based on the term which is zero. We are using the fact that the sequence is decidable (in the sense that each element is either zero or one). (Contributed by Jim Kingdon, 23-Aug-2023.) |
Theorem | trilpolemlt1 13455* | Lemma for trilpo 13457. The case. We can use the distance between and one (that is, ) to find a position in the sequence where terms after that point will not add up to as much as . By finomni 7033 we know the terms up to either contain a zero or are all one. But if they are all one that contradicts the way we constructed , so we know that the sequence contains a zero. (Contributed by Jim Kingdon, 23-Aug-2023.) |
Theorem | trilpolemres 13456* | Lemma for trilpo 13457. The result. (Contributed by Jim Kingdon, 23-Aug-2023.) |
Theorem | trilpo 13457* |
Real number trichotomy implies the Limited Principle of Omniscience
(LPO). We expect that we'd need some form of countable choice to prove
the converse.
Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence contains a zero or it is all ones. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. Compare it with one using trichotomy. The three cases from trichotomy are trilpolemlt1 13455 (which means the sequence contains a zero), trilpolemeq1 13454 (which means the sequence is all ones), and trilpolemgt1 13453 (which is not possible). Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 13443) or that the real numbers are a discrete field (see trirec0 13458). LPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qtri3or 10080 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.) |
Omni | ||
Theorem | trirec0 13458* |
Every real number having a reciprocal or equaling zero is equivalent to
real number trichotomy.
This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 13457). (Contributed by Jim Kingdon, 10-Jun-2024.) |
Theorem | trirec0xor 13459* |
Version of trirec0 13458 with exclusive-or.
The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.) |
Theorem | apdifflemf 13460 | Lemma for apdiff 13462. Being apart from the point halfway between and suffices for to be a different distance from and from . (Contributed by Jim Kingdon, 18-May-2024.) |
# # | ||
Theorem | apdifflemr 13461 | Lemma for apdiff 13462. (Contributed by Jim Kingdon, 19-May-2024.) |
# # # | ||
Theorem | apdiff 13462* | The irrationals (reals apart from any rational) are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 17-May-2024.) |
# # | ||
Theorem | iswomninnlem 13463* | Lemma for iswomnimap 7059. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.) |
frec WOmni DECID | ||
Theorem | iswomninn 13464* | Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7059 but it will sometimes be more convenient to use and rather than and . (Contributed by Jim Kingdon, 20-Jun-2024.) |
WOmni DECID | ||
Theorem | iswomni0 13465* | Weak omniscience stated in terms of equality with . Like iswomninn 13464 but with zero in place of one. (Contributed by Jim Kingdon, 24-Jul-2024.) |
WOmni DECID | ||
Theorem | ismkvnnlem 13466* | Lemma for ismkvnn 13467. The result, with a hypothesis to give a name to an expression for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.) |
frec Markov | ||
Theorem | ismkvnn 13467* | The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.) |
Markov | ||
Theorem | redcwlpolemeq1 13468* | Lemma for redcwlpo 13469. A biconditionalized version of trilpolemeq1 13454. (Contributed by Jim Kingdon, 21-Jun-2024.) |
Theorem | redcwlpo 13469* |
Decidability of real number equality implies the Weak Limited Principle
of Omniscience (WLPO). We expect that we'd need some form of countable
choice to prove the converse.
Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 13468). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones. Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO". WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10084 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.) |
DECID WOmni | ||
Theorem | tridceq 13470* | Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 13457 and redcwlpo 13469). Thus, this is an analytic analogue to lpowlpo 7061. (Contributed by Jim Kingdon, 24-Jul-2024.) |
DECID | ||
Theorem | redc0 13471* | Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.) |
DECID DECID | ||
Theorem | reap0 13472* | Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.) |
DECID # | ||
Theorem | dceqnconst 13473* | Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 13469 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.) |
DECID | ||
Theorem | dcapnconst 13474* |
Decidability of real number apartness implies the existence of a certain
non-constant function from real numbers to integers. Variation of
Exercise 11.6(i) of [HoTT], p. (varies).
See trilpo 13457 for more
discussion of decidability of real number apartness.
This is a weaker form of dceqnconst 13473 and in fact this theorem can be proved using dceqnconst 13473 as shown at dcapnconstALT 13475. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) |
DECID # | ||
Theorem | dcapnconstALT 13475* | Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 13474 by means of dceqnconst 13473. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
DECID # | ||
Theorem | nconstwlpolem0 13476* | Lemma for nconstwlpo 13479. If all the terms of the series are zero, so is their sum. (Contributed by Jim Kingdon, 26-Jul-2024.) |
Theorem | nconstwlpolemgt0 13477* | Lemma for nconstwlpo 13479. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.) |
Theorem | nconstwlpolem 13478* | Lemma for nconstwlpo 13479. (Contributed by Jim Kingdon, 23-Jul-2024.) |
Theorem | nconstwlpo 13479* | Existence of a certain non-constant function from reals to integers implies WOmni (the Weak Limited Principle of Omniscience or WLPO). Based on Exercise 11.6(ii) of [HoTT], p. (varies). (Contributed by BJ and Jim Kingdon, 22-Jul-2024.) |
WOmni | ||
Theorem | neapmkvlem 13480* | Lemma for neapmkv 13481. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.) |
# | ||
Theorem | neapmkv 13481* | If negated equality for real numbers implies apartness, Markov's Principle follows. Exercise 11.10 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Jun-2024.) |
# Markov | ||
Theorem | supfz 13482 | The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.) |
Theorem | inffz 13483 | The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.) |
inf | ||
Theorem | taupi 13484 | Relationship between and . This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.) |
Theorem | ax1hfs 13485 | Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.) |
Theorem | dftest 13486 |
A proposition is testable iff its negative or double-negative is true.
See Chapter 2 [Moschovakis] p. 2.
We do not formally define testability with a new token, but instead use DECID before the formula in question. For example, DECID corresponds to " is testable". (Contributed by David A. Wheeler, 13-Aug-2018.) For statements about testable propositions, search for the keyword "testable" in the comments of statements, for instance using the Metamath command "MM> SEARCH * "testable" / COMMENTS". (New usage is discouraged.) |
DECID | ||
These are definitions and proofs involving an experimental "allsome" quantifier (aka "all some"). In informal language, statements like "All Martians are green" imply that there is at least one Martian. But it's easy to mistranslate informal language into formal notations because similar statements like do not imply that is ever true, leading to vacuous truths. Some systems include a mechanism to counter this, e.g., PVS allows types to be appended with "+" to declare that they are nonempty. This section presents a different solution to the same problem. The "allsome" quantifier expressly includes the notion of both "all" and "there exists at least one" (aka some), and is defined to make it easier to more directly express both notions. The hope is that if a quantifier more directly expresses this concept, it will be used instead and reduce the risk of creating formal expressions that look okay but in fact are mistranslations. The term "allsome" was chosen because it's short, easy to say, and clearly hints at the two concepts it combines. I do not expect this to be used much in metamath, because in metamath there's a general policy of avoiding the use of new definitions unless there are very strong reasons to do so. Instead, my goal is to rigorously define this quantifier and demonstrate a few basic properties of it. The syntax allows two forms that look like they would be problematic, but they are fine. When applied to a top-level implication we allow ! , and when restricted (applied to a class) we allow ! . The first symbol after the setvar variable must always be if it is the form applied to a class, and since cannot begin a wff, it is unambiguous. The looks like it would be a problem because or might include implications, but any implication arrow within any wff must be surrounded by parentheses, so only the implication arrow of ! can follow the wff. The implication syntax would work fine without the parentheses, but I added the parentheses because it makes things clearer inside larger complex expressions, and it's also more consistent with the rest of the syntax. For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html I hope that others will eventually agree that allsome is awesome. | ||
Syntax | walsi 13487 | Extend wff definition to include "all some" applied to a top-level implication, which means is true whenever is true, and there is at least least one where is true. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Syntax | walsc 13488 | Extend wff definition to include "all some" applied to a class, which means is true for all in , and there is at least one in . (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Definition | df-alsi 13489 | Define "all some" applied to a top-level implication, which means is true whenever is true and there is at least one where is true. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Definition | df-alsc 13490 | Define "all some" applied to a class, which means is true for all in and there is at least one in . (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Theorem | alsconv 13491 | There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.) |
! ! | ||
Theorem | alsi1d 13492 | Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Theorem | alsi2d 13493 | Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Theorem | alsc1d 13494 | Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Theorem | alsc2d 13495 | Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! |
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