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**Theorem List for Intuitionistic Logic Explorer -** 15501-15579 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	peano4nninf 15501*	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 15502*	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	nninfalllem1 15503*	Lemma for nninfall 15504. (Contributed by Jim Kingdon, 1-Aug-2022.)
ℕ_∞ ℕ_∞

Theorem	nninfall 15504*	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	nninfsellemdc 15505*	Lemma for nninfself 15508. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.)
ℕ_∞ $/\$ DECID

Theorem	nninfsellemcl 15506*	Lemma for nninfself 15508. (Contributed by Jim Kingdon, 8-Aug-2022.)
ℕ_∞ $/\$

Theorem	nninfsellemsuc 15507*	Lemma for nninfself 15508. (Contributed by Jim Kingdon, 6-Aug-2022.)
ℕ_∞ $/\$

Theorem	nninfself 15508*	Domain and range of the selection function for ℕ_∞. (Contributed by Jim Kingdon, 6-Aug-2022.)
ℕ_∞ ℕ_∞ℕ_∞

Theorem	nninfsellemeq 15509*	Lemma for nninfsel 15512. (Contributed by Jim Kingdon, 9-Aug-2022.)
ℕ_∞ ℕ_∞

Theorem	nninfsellemqall 15510*	Lemma for nninfsel 15512. (Contributed by Jim Kingdon, 9-Aug-2022.)
ℕ_∞ ℕ_∞

Theorem	nninfsellemeqinf 15511*	Lemma for nninfsel 15512. (Contributed by Jim Kingdon, 9-Aug-2022.)
ℕ_∞ ℕ_∞

Theorem	nninfsel 15512*	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 15513*	Lemma for nninfomni 15514. (Contributed by Jim Kingdon, 10-Aug-2022.)
ℕ_∞ ℕ_∞ Omni

Theorem	nninfomni 15514	ℕ_∞ is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.)
ℕ_∞ Omni

Theorem	nninffeq 15515*	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	nnnninfen 15516	Equinumerosity of the natural numbers and ℕ_∞ is equivalent to the Limited Principle of Omniscience (LPO). Remark in Section 1.1 of [Pradic2025], p. 2. (Contributed by Jim Kingdon, 8-Jul-2025.)
ℕ_∞ Omni

13.3.5 Schroeder-Bernstein Theorem

Theorem	exmidsbthrlem 15517*	Lemma for exmidsbthr 15518. (Contributed by Jim Kingdon, 11-Aug-2022.)
ℕ_∞ $/\$ EXMID

Theorem	exmidsbthr 15518*	The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.)
$/\$ EXMID

Theorem	exmidsbth 15519*	The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 7026) 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-intuitionistic proof at https://us.metamath.org/mpeuni/sbth.html 7026. The reverse direction (exmidsbthr 15518) 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 15520	Lemma for sbthom 15521. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.)
Omni ⊔ $\/$

Theorem	sbthom 15521	Schroeder-Bernstein is not possible even for . We know by exmidsbth 15519 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

13.3.6 Real and complex numbers

Theorem	qdencn 15522*	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 11346 (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 15523*	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 15524	Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.)
$/\$ $\/$ $\/$ DECID #

Theorem	isomninnlem 15525*	Lemma for isomninn 15526. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.)
frec Omni $\/$

Theorem	isomninn 15526*	Omniscience stated in terms of natural numbers. Similar to isomnimap 7196 but it will sometimes be more convenient to use and rather than and . (Contributed by Jim Kingdon, 30-Aug-2023.)
Omni $\/$

Theorem	cvgcmp2nlemabs 15527*	Lemma for cvgcmp2n 15528. 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 15528*	A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.)
$/\$ $/\$ $/\$

Theorem	iooref1o 15529	A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.)


Theorem	iooreen 15530	An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)


13.3.7 Analytic omniscience principles 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 7201), (1) the Limited Principle of Omniscience (LPO) is Omni (see df-omni 7194), (2) the Weak Limited Principle of Omniscience (WLPO) is WOmni (see df-womni 7223), (3) Markov's Principle (MP) is Markov (see df-markov 7211), (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 15538), (2) Analytic WLPO is decidability of real number equality, DECID (see redcwlpo 15550), (3) Analytic MP is # (see neapmkv 15563), (4) Analytic LLPO is real number dichotomy, $\/$ (most relevant current theorem is maxclpr 11366).

Theorem	trilpolemclim 15531*	Lemma for trilpo 15538. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.)


Theorem	trilpolemcl 15532*	Lemma for trilpo 15538. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.)


Theorem	trilpolemisumle 15533*	Lemma for trilpo 15538. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.)


Theorem	trilpolemgt1 15534*	Lemma for trilpo 15538. The case. (Contributed by Jim Kingdon, 23-Aug-2023.)


Theorem	trilpolemeq1 15535*	Lemma for trilpo 15538. 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 15536*	Lemma for trilpo 15538. 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 7199 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 15537*	Lemma for trilpo 15538. The result. (Contributed by Jim Kingdon, 23-Aug-2023.)
$\/$ $\/$ $\/$

Theorem	trilpo 15538*	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 15536 (which means the sequence contains a zero), trilpolemeq1 15535 (which means the sequence is all ones), and trilpolemgt1 15534 (which is not possible). Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 15524) or that the real numbers are a discrete field (see trirec0 15539). 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 10310 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.)
$\/$ $\/$ Omni

Theorem	trirec0 15539*	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 15538). (Contributed by Jim Kingdon, 10-Jun-2024.)
$\/$ $\/$ $\/$

Theorem	trirec0xor 15540*	Version of trirec0 15539 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 15541	Lemma for apdiff 15543. 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 15542	Lemma for apdiff 15543. (Contributed by Jim Kingdon, 19-May-2024.)
# $/\$ # #

Theorem	apdiff 15543*	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 15544*	Lemma for iswomnimap 7225. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.)
frec WOmni DECID

Theorem	iswomninn 15545*	Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7225 but it will sometimes be more convenient to use and rather than and . (Contributed by Jim Kingdon, 20-Jun-2024.)
WOmni DECID

Theorem	iswomni0 15546*	Weak omniscience stated in terms of equality with . Like iswomninn 15545 but with zero in place of one. (Contributed by Jim Kingdon, 24-Jul-2024.)
WOmni DECID

Theorem	ismkvnnlem 15547*	Lemma for ismkvnn 15548. 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 15548*	The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.)
Markov

Theorem	redcwlpolemeq1 15549*	Lemma for redcwlpo 15550. A biconditionalized version of trilpolemeq1 15535. (Contributed by Jim Kingdon, 21-Jun-2024.)


Theorem	redcwlpo 15550*	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 15549). 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 10314 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)
DECID WOmni

Theorem	tridceq 15551*	Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 15538 and redcwlpo 15550). Thus, this is an analytic analogue to lpowlpo 7227. (Contributed by Jim Kingdon, 24-Jul-2024.)
$\/$ $\/$ DECID

Theorem	redc0 15552*	Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.)
DECID DECID

Theorem	reap0 15553*	Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.)
$\/$ $\/$ DECID #

Theorem	cndcap 15554*	Real number trichotomy is equivalent to decidability of complex number apartness. (Contributed by Jim Kingdon, 10-Apr-2025.)
$\/$ $\/$ DECID #

Theorem	dceqnconst 15555*	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 15550 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 15556*	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 15538 for more discussion of decidability of real number apartness. This is a weaker form of dceqnconst 15555 and in fact this theorem can be proved using dceqnconst 15555 as shown at dcapnconstALT 15557. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.)
DECID # $/\$ $/\$

Theorem	dcapnconstALT 15557*	Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 15556 by means of dceqnconst 15555. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
DECID # $/\$ $/\$

Theorem	nconstwlpolem0 15558*	Lemma for nconstwlpo 15561. If all the terms of the series are zero, so is their sum. (Contributed by Jim Kingdon, 26-Jul-2024.)


Theorem	nconstwlpolemgt0 15559*	Lemma for nconstwlpo 15561. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.)


Theorem	nconstwlpolem 15560*	Lemma for nconstwlpo 15561. (Contributed by Jim Kingdon, 23-Jul-2024.)
$/\$ $\/$

Theorem	nconstwlpo 15561*	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 15562*	Lemma for neapmkv 15563. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
$/\$ #

Theorem	neapmkv 15563*	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	neap0mkv 15564*	The analytic Markov principle can be expressed either with two arbitrary real numbers, or one arbitrary number and zero. (Contributed by Jim Kingdon, 23-Feb-2025.)
# #

Theorem	ltlenmkv 15565*	If can be expressed as holding exactly when holds and the values are not equal, then the analytic Markov's Principle applies. (To get the regular Markov's Principle, combine with neapmkv 15563). (Contributed by Jim Kingdon, 23-Feb-2025.)
$/\$ #

13.3.8 Supremum and infimum

Theorem	supfz 15566	The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)


Theorem	inffz 15567	The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.)
inf

13.3.9 Circle constant

Theorem	taupi 15568	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.)


13.4 Mathbox for Mykola Mostovenko

Theorem	ax1hfs 15569	Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.)
$/\$

13.5 Mathbox for David A. Wheeler

13.5.1 Testable propositions

Theorem	dftest 15570	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 $\/$

13.5.2 Allsome quantifier 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 15571	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 15572	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 15573	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 15574	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 15575	There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.)
! !

Theorem	alsi1d 15576	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 15577	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 15578	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 15579	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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