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Created by Mario Carneiro

Intuitionistic Logic Proof Explorer

Intuitionistic Logic (Wikipedia [accessed 19-Jul-2015], Stanford Encyclopedia of Philosophy [accessed 19-Jul-2015]) can be thought of as a constructive logic in which we must build and exhibit concrete examples of objects before we can accept their existence. Unproved statements in intuitionistic logic are not given an intermediate truth value, instead, they remain of unknown truth value until they are either proved or disproved. Intuitionist logic can also be thought of as a weakening of classical logic such that the law of excluded middle (LEM), (φ ¬ φ), doesn't always hold. Specifically, it holds if we have a proof for φ or we have a proof for ¬ φ, but it doesn't necessarily hold if we don't have a proof of either one. There is also no rule for double negation elimination. Brouwer observed in 1908 that LEM was abstracted from finite situations, then extended without justification to statements about infinite collections.

 Contents of this page Overview of intuitionistic logic Overview of this work The axioms Some theorems How to intuitionize classical proofs Metamath Proof Explorer cross reference Bibliography Related pages Table of Contents and Theorem List Most Recent Proofs (this mirror) (latest) Bibliographic Cross-Reference Definition List ASCII Equivalents for Text-Only Browsers Metamath database iset.mm (ASCII file) External links GitHub repository [accessed 06-Jan-2018]

Overview of intuitionistic logic

(Placeholder for future use)

Overview of this work

(By Gérard Lang, 7-May-2018)

Mario Carneiro's work (Metamath database) "iset.mm" provides in Metamath a development of "set.mm" whose eventual aim is to show how many of the theorems of set theory and mathematics that can be derived from classical first order logic can also be derived from a weaker system called "intuitionistic logic." To achieve this task, iset.mm adds (or substitutes) intuitionistic axioms for a number of the classical logical axioms of set.mm.

Among these new axioms, the first six (ax-ia1, ax-ia2, ax-ia3, ax-io, ax-in1, and ax-in2), when added to ax-1, ax-2, and ax-mp, allow for the development of intuitionistic propositional logic. We omit the classical axiom ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)) (which is ax-3 in set.mm). Each of our new axioms is a theorem of classical propositional logic, but ax-3 cannot be derived from them. Similarly, other basic classical theorems, like the third middle excluded or the equivalence of a proposition with its double negation, cannot be derived in intuitionistic propositional calculus. Glivenko showed that a proposition φ is a theorem of classical propositional calculus if and only if ¬¬φ is a theorem of intuitionistic propositional calculus.

The next 4 new axioms (ax-ial, ax-i5r, ax-ie1, and ax-ie2) together with the set.mm axioms ax-4, ax-5, ax-7, and ax-gen do not mention equality or distinct variables.

The ax-i9 axiom is just a slight variation of the classical ax-9. The classical axiom ax-12 is strengthened into first ax-i12 and then ax-bndl (two results which would be fairly readily equivalent to ax-12 classically but which do not follow from ax-12, at least not in an obvious way, in intuitionistic logic). The substitution of ax-i9, ax-i12, and ax-bndl for ax-9 and ax-12 and the inclusion of ax-8, ax-10, ax-11, ax-13, ax-14, and ax-17 allow for the development of the intuitionistic predicate calculus.

Each of the new axioms is a theorem of classical first order logic with equality. But some axioms of classical first order logic with equality, like ax-3, cannot be derived in the intuitionistic predicate calculus.

One of the major interests of the intuitionistic predicate calculus is that its use can be considered as a realization of the program of the constructivist philosophical view of mathematics.

The axioms

As with the classical axioms we have propositional logic and predicate logic.

The axioms of intuitionistic propositional logic consist of some of the axioms from classical propositional logic, plus additional axioms for the operation of the 'and', 'or' and 'not' connectives.

 Axiom Simp ax-1 ⊢ (φ → (ψ → φ)) Axiom Frege ax-2 ⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ))) Rule of Modus Ponens ax-mp ⊢ 𝜑   &   ⊢ 𝜑 → 𝜓   ⇒   ⊢ 𝜓 Left 'and' elimination ax-ia1 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) Right 'and' elimination ax-ia2 ⊢ ((φ ∧ ψ) → ψ) 'And' introduction ax-ia3 ⊢ (φ → (ψ → (φ ∧ ψ))) Definition of 'or' ax-io ⊢ (((φ ∨ χ) → ψ) ↔ ((φ → ψ) ∧ (χ → ψ))) 'Not' introduction ax-in1 ⊢ ((φ → ¬ φ) → ¬ φ) 'Not' elimination ax-in2 ⊢ (¬ φ → (φ → ψ))

Unlike in classical propositional logic, 'and' and 'or' are not readily defined in terms of implication and 'not'. In particular, φψ is not equivalent to ¬ φψ, nor is φψ equivalent to ¬ φψ, nor is φψ equivalent to ¬ (φ → ¬ ψ).

The ax-in1 axiom is a form of proof by contradiction which does hold intuitionistically. That is, if φ implies a contradiction (such as its own negation), then one can conclude ¬ φ. By contrast, assuming ¬ φ and then deriving a contradiction only serves to prove ¬ ¬ φ, which in intuitionistic logic is not the same as φ.

The biconditional can be defined as the conjunction of two implications, as in dfbi2 and df-bi.

Predicate logic adds set variables (individual variables) and the ability to quantify them with ∀ (for-all) and ∃ (there-exists). Unlike in classical logic, ∃ cannot be defined in terms of ∀. As in classical logic, we also add = for equality (which is key to how we handle substitution in metamath) and ∈ (which for current purposes can just be thought of as an arbitrary predicate, but which will later come to mean set membership).

Our axioms are based on the classical set.mm predicate logic axioms, but adapted for intuitionistic logic, chiefly by adding additional axioms for ∃ and also changing some aspects of how we handle negations.

 Axiom of Specialization ax-4 ⊢ (∀xφ → φ) Axiom of Quantified Implication ax-5 ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ)) The converse of ax-5o ax-i5r ⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ψ)) Axiom of Quantifier Commutation ax-7 ⊢ (∀x∀yφ → ∀y∀xφ) Rule of Generalization ax-gen ⊢ φ   =>    ⊢ ∀xφ x is bound in ∀xφ ax-ial ⊢ (∀xφ → ∀x∀xφ) x is bound in ∃xφ ax-ie1 ⊢ (∃xφ → ∀x∃xφ) Define existential quantification ax-ie2 ⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ))) Axiom of Equality ax-8 ⊢ (x = y → (x = z → y = z)) Axiom of Existence ax-i9 ⊢ ∃x x = y Axiom of Quantifier Substitution ax-10 ⊢ (∀x x = y → ∀y y = x) Axiom of Variable Substitution ax-11 ⊢ (x = y → (∀yφ → ∀x(x = y → φ))) Axiom of Quantifier Introduction ax-i12 ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) Axiom of Bundling ax-bndl ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀x∀z(x = y → ∀z x = y))) Left Membership Equality ax-13 ⊢ (x = y → (x ∈ z → y ∈ z)) Right Membership Equality ax-14 ⊢ (x = y → (z ∈ x → z ∈ y)) Distinctness ax-17 ⊢ (φ → ∀xφ), where x does not occur in φ

Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets." A set can be an element of another set, and this relationship is indicated by the symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects. These axioms were developed in response to Russell's Paradox, a discovery that revolutionized the foundations of mathematics and logic.

In the IZF axioms that follow, all set variables are assumed to be distinct. If you click on their links you will see the explicit distinct variable conditions.

 Axiom of Extensionality ax-ext ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y) Axiom of Collection ax-coll ⊢ (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀x ∈ 𝑎 ∃y ∈ 𝑏 φ) Axiom of Separation ax-sep ⊢ ∃y∀x(x ∈ y ↔ (x ∈ z ∧ φ)) Axiom of Power Sets ax-pow ⊢ ∃y∀z(∀w(w ∈ z → w ∈ x) → z ∈ y) Axiom of Pairing ax-pr ⊢ ∃z∀w((w = x ∨ w = y) → w ∈ z) Axiom of Union ax-un ⊢ ∃y∀z(∃w(z ∈ w ∧ w ∈ x) → z ∈ y) Axiom of Set Induction ax-setind ⊢ (∀𝑎(∀y ∈ 𝑎 [y / 𝑎]φ → φ) → ∀𝑎φ) Axiom of Infinity ax-iinf ⊢ ∃x(∅ ∈ x ∧ ∀y(y ∈ x → suc y ∈ x))

We develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. The one exception to the statement that we use IZF is that a few sections develop set theory using Constructive Zermelo-Fraenkel (CZF), also described in Crosilla. These sections start at wbd (including the section header right before it) and the biggest complication is the machinery to classify formulas as bounded formulas, for purposes of the Axiom of Restricted Separation ax-bdsep.

A Theorem Sampler

From a psychological point of view, learning constructive mathematics is agonizing, for it requires one to first unlearn certain deeply ingrained intuitions and habits acquired during classical mathematical training.
—Andrej Bauer

Listed here are some examples of starting points for your journey through the maze. Some are stated just as they would be in a non-constructive context; others are here to highlight areas which look different constructively. You should study some simple proofs from propositional calculus until you get the hang of it. Then try some predicate calculus and finally set theory.

The Theorem List shows the complete set of theorems in the database. You may also find the Bibliographic Cross-Reference useful.

 Propositional calculus Law of identity idALT Praeclarum theorema anim12 Contraposition introduction con3 Double negation introduction notnot Triple negation notnotnot Definition of exclusive or df-xor Negation and the false constant dfnot Predicate calculus Existential and universal quantifier swap 19.12 Existentially quantified implication 19.35-1 x = x equid Definition of proper substitution df-sb Double existential uniqueness 2eu7 Set theory Commutative law for union uncom A basic relationship between class and wff variables abeq2 Two ways of saying "is a set" isset The ZF axiom of foundation implies excluded middle regexmid Russell's paradox ru Ordinal trichotomy implies excluded middle ordtriexmid Mathematical (finite) induction findes Peano's postulates for arithmetic: peano1 peano2 peano3 peano4 peano5 Two natural numbers are either equal or not equal nndceq (a special case of the law of the excluded middle which we can prove). A natural number is either zero or a successor nn0suc The axiom of choice implies excluded middle acexmid Burali-Forti paradox onprc Transfinite induction tfis3 Closure law for ordinal addition oacl Real and complex numbers Archimedean property of real numbers arch Properties of apartness: apirr apsym apcotr apti The square root of 2 is irrational sqrt2irrap (a different statement than "The square root of 2 is not rational" sqrt2irr) Convergence of a sequence of complex numbers climcvg1n given a condition on the rate of convergence Triangle inequality for absolute value abstrii The maximum of two real numbers maxleb

How to intuitionize classical proofs

For the student or master of classical mathematics, constructive mathematics can be baffling. One can get over some of the intial hurdles of understanding how constructive mathematics works and why it might be interesting by reading [Bauer] but that work does little to explain in concrete terms how to write proofs in intuitionistic logic. Fortunately, metamath helps with one of the biggest hurdles: noticing when one is even using the law of the excluded middle or the axiom of choice. But suppose you have a classical proof from the Metamath Proof Explorer and it fails to verify when you copy it over to the Intuitionistic Logic Explorer. What then? Here are some rules of thumb in converting classical proofs to intuitionistic ones.

• If you see case elimination ( pm2.61 or its variants) you'll probably end up with two theorems for the two cases. In particular, if the cases were 𝐴 ∈ V and ¬ 𝐴 ∈ V you probably just care about the 𝐴 ∈ V case. On the other hand, if the proposition being eliminated is decidable (for example due to nndceq, zdceq, zdcle, zdclt, eluzdc, or fzdcel), then case elimination will work using theorems such as df-dc and mpjaodan.
• Non-empty almost always needs to be changed to inhabited (those terms are defined at n0rf).
• If the original proof relied on propositional/predicate logic which isn't a theorem of intuitionistic logic, maybe there is a way of expressing the logic more directly. This is perhaps the hardest one to put in cookbook form: you might need to try some things and see if anything works.
• If the original proof relied on df-ex so that it could prove a theorem for and then get for free (or vice versa), instead go look at the original proof and try to come up the analogues to each step for the other quantifier (for example, cbvrexcsf, sbcrext, rexxpf). Similarly, if you have a theorem for and are trying to prove the corresponding theorem for < you'll probably need to use analogous steps rather than negation (examples: leaddsub, ltsub1, ltsub2).
• If you are dealing with a definition, try to find the best constructive definition from the literature ([HoTT] book, Stanford Encyclopedia of Philosophy, [Bauer], etc). Once you pick a definition, that'll affect the proofs which rely on that definition.
• If there is case elimination on whether variables are distinct, most of the time you just need the variant with distinct variables. Sometimes you can then remove the constraint with a temporary variable (e.g. the various sbco2* variants, nfralya, r19.3rm).
• Sometimes one direction of a biconditional holds, or subset holds instead of equality. You might be able to keep the easy direction and worry about the other one later.
• If there is case elimination sometimes only one of the two cases is possible. For example, in mosubopt the rest of the formula being proved constrains which case matters.
• If you need an additional condition (for example, because the original proof used case elimination) and you are proving a biconditional, consider whether both sides of the biconditional imply the condition. If so, you'll be able to prove the biconditional with that condition as an antecedent, and then use pm5.21nii or one of its variants to remove the antecedent (example: elxp4).
• If your proof relies on dveeq2 try dveeq2or and likewise for the other things downstream of ax-i12 or ax-bndl.
• If you have a disjunction, be reluctant to turn it into an implication using ord and the like. Instead, show that each disjunct implies what you are trying to prove and use jaoi to join those two implications into something which can hook up to the disjunction.
• Disjunctive syllogism holds in intuitionistic logic and we state it a few ways (for example orel1 and ecased) but we don't have a wide variety of convenience theorems. Unless we add those, you'll use ord or something similar followed by mpd or something similar. This may add a few steps but they are straightforward ones.
• If your proof is doing tricky things perhaps in the interest of shortness, try just expanding the definitions and applying logic in a straightforward way. See if this gets you a working (although perhaps longer) proof.
• If your proof relies on a biconditional in set.mm which isn't in iset.mm, see if one direction is in iset.mm and see which direction your proof is using. For example 19.35-1 or exnalim.
• If you are doing things with inhabited classes (beyond just applying existing inhabited class theorems), you may be able to dig up some predicate logic you haven't used in a while (e.g. raaan).
• Consider the possibility of giving up. Some things just won't have intuitionistic proofs. The more it looks like excluded middle or other non-intuitionistic statements, the more likely you are dealing with one of these. But it can be hard to have good intuition about this. In some cases it may be possible to ask "can I use this statement to prove 𝜑 ∨ ¬ 𝜑 for an arbitrary proposition" (see ordtriexmid for example ), but this is not always an easy technique to apply.
• Switching between 2th and 2false might help (e.g. dfnul2, dfnul3, rab0).
• In many cases statements which are equivalent in classical logic become several non-equivalent statements (e.g. exclusive or, ordinals, non-empty versus inhabited, apartness versus negated equality). This is usually a good place to look for a literature reference, but don't be afraid to change the statement being proved to "what you really meant is X" as appropriate.
• If a statement has multiple equivalences in set.mm (e.g. mof and mo3 , or dffun2 and dffun3 ) and only some of them in iset.mm, sometimes a pretty similar proof will work (that is, which one to use in the original proof may have been a fairly arbitrary choice).
• A number of theorems related to functions (especially ovex and fvex ) in set.mm perform case elimination based on whether we are evaluating the function within its domain or outside it. The most straightforward solution is to use fnovex or funfvex which only work within the domain. Using these may involve rearranging logic, for example by changing rexlimivw to rexlimdva (example: ovelrn or indeed most uses of fnovex and funfvex). If a function value is inhabited, we know we are evaluating it within its domain by relelfvdm.
• With excluded middle, ∅ ∈ 𝐴 and 𝐴 ≠ ∅ are equivalent (where 𝐴 is an ordinal). In such theorems, ∅ ∈ 𝐴 is generally the more interesting condition constructively.
• Reverse closure in set.mm uses excluded middle (for example ovrcl or ndmfvrcl ). The most general way to handle this is to add more conditions that we are evaluating operations within their domains (for example set.mm's addasspi versus iset.mm's addasspig in which conditions such as 𝐴N are added, or set.mm's ltbtwnnq versus iset.mm's ltbtwnnqq, in which 𝑥 is changed to 𝑥Q). If the result of applying a function is inhabited, then we know we applied it within its domain - that is relelfvdm or elmpocl may be useful. (These thoughts apply to operations - in at least one place, dvdszrcl, set.mm uses the term "reverse closure" for binary relations - but this is a lot more like brel than like the reverse closure theorems described above).
• With excluded middle not equal () and apart (#) are equivalent. When working with real and complex numbers, apartness is almost always what you want. See df-ap for more on apartness.
• Given a theorem of the form 𝑋 = 𝑌𝑍 = 𝑊 we can derive 𝑋𝑌𝑍𝑊 but in many contexts what we really want is 𝑋 # 𝑌𝑍 # 𝑊. See if the version with apartness already exists and if not consider adding it (building from basic apartness theorems like apadd1 and apmul1 for example). Once you have proved the version using apartness you can use it to prove the version with equality if you don't already have it, using notbid and apti.
• Exclusive or (𝜑𝜓) is equivalent to 𝜑 ↔ ¬ 𝜓 given excluded middle but we just have one direction (xorbin). Consider intuitionizing 𝜑 ↔ ¬ 𝜓 as 𝜑𝜓 (example: rpnegap).
• The expression ( I ‘𝐴) evaluates to 𝐴 if 𝐴 is a set (by fvi) and the empty set if 𝐴 is a proper class (by fvprc). However, combining these two facts and thus using I to protect against proper classes does not seem to be possible without excluded middle (set.mm examples: sumeq2ii and strfvi ). Usually you will end up adding a 𝐴 ∈ V condition in such cases.
• If you get stuck, ask! (for example in a GitHub issue or on the mailing list). We have a number of contributors who have experience in constructive mathematics in general, or iset.mm in particular, and one of the best things about doing/learning mathematics in metamath is the collaborative nature of how we develop it.

Metamath Proof Explorer cross reference

This is a list of theorems from the Metamath Proof Explorer (which assumes the law of the excluded middle throughout) which we do not have in the Intuitionistic Logic Explorer (generally because they are not provable without the law of the excluded middle, although some could be proved but aren't for a variety of reasons), together with the closest replacements.

set.mm iset.mm notes
ax-3 , con4d , con34b , necon4d , necon4bd con3 The form of contraposition which removes negation does not hold in intutionistic logic.
pm2.18 pm2.01 See for example [Bauer] who uses the terminology "proof of negation" versus "proof by contradiction" to distinguish these.
pm2.18d , pm2.18i pm2.01d See for example [Bauer] who uses the terminology "proof of negation" versus "proof by contradiction" to distinguish these.
notnotrd , notnotri , notnotr , notnotb notnot Double negation introduction holds but not double negation elimination.
mt3d mtod
nsyl2 nsyl
mt4d mt2d
nsyl4 con1dc
pm2.61 , pm2.61d , pm2.61d1 , pm2.61d2 , pm2.61i , pm2.61ii , pm2.61nii , pm2.61iii , pm2.61ian , pm2.61dan , pm2.61ddan , pm2.61dda , pm2.61ine , pm2.61ne , pm2.61dne , pm2.61dane , pm2.61da2ne , pm2.61da3ne , pm2.61iine none If the proposition being eliminated is decidable (for example due to nndceq, zdceq, zdcle, zdclt, eluzdc, or fzdcel), then case elimination will work using theorems such as exmiddc and mpjaodan
dfbi1 , dfbi3 df-bi, dfbi2
impcon4bid, con4bid, notbi, con1bii, con4bii, con2bii con3, condc
xor3 , nbbn xorbin, xornbi, xor3dc, nbbndc
biass biassdc
df-or , pm4.64 , pm2.54 , orri , orrd pm2.53, ori, ord, dfordc
imor , imori imorr, imorri, imordc
pm4.63 pm3.2im
iman imanim, imanst
annim annimim
oran , pm4.57 oranim, orandc
ianor pm3.14, ianordc
pm4.14 pm4.14dc, pm3.37
pm4.52 pm4.52im
pm4.53 pm4.53r
pm5.17 xorbin The combination of df-xor and xorbin is the forward direction of pm5.17
biluk bilukdc
ecase none This is a form of case elimination.
ecase3d none This is a form of case elimination.
dedlem0b dedlemb
pm4.42 pm4.42r
3ianor 3ianorr
df-nan and other theorems using NAND (Sheffer stroke) notation none A quick glance at the internet shows this mostly being used in the presence of excluded middle; in any case it is not currently present in iset.mm
df-xor df-xor Although the definition of exclusive or is called df-xor in both set.mm and iset.mm (at least currently), the definitions are not equivalent (in the absence of excluded middle).
xnor none The set.mm proof uses theorems not in iset.mm.
xorass none The set.mm proof uses theorems not in iset.mm.
xor2 xoranor, xor2dc
xornan xor2dc
xornan2 none See discussion under df-nan
xorneg2 , xorneg1 , xorneg none The set.mm proofs use theorems not in iset.mm.
xorexmid none A form of excluded middle
df-ex exalim
alex alexim
exnal exnalim
alexn alexnim
exanali exanaliim
19.35 , 19.35ri 19.35-1
19.30 none
19.39 i19.39
19.24 i19.24
19.36 , 19.36v 19.36-1
19.37 , 19.37v 19.37-1
19.32 19.32r
19.31 19.31r
exdistrf exdistrfor
df-mo df-mo The definitions are different although they currently share the same name.
exmo exmonim
mof mo2r, mo3
df-eu , dfeu eu5 Although this is a definition in set.mm and a theorem in iset.mm it is otherwise the same (with a different name).
eu6 df-eu Although this is a definition in iset.mm and a theorem in set.mm it is otherwise the same (with a different name).
nfabd2 nfabd
nne nner, nnedc
exmidne dcne
necon1bd necon1bddc
necon4bd necon4bddc
necon1d necon1ddc
necon4d necon4ddc
necon1ai necon1aidc
necon1bi necon1bidc
necon4ai necon4aidc
necon1i necon1idc
necon4i necon4idc
necon4abid necon4abiddc
necon4bbid necon4bbiddc
necon4bid necon4biddc, apcon4bid
necon1abii necon1abiidc
necon1bbii necon1bbiidc
necon1abid necon1abiddc
necon1bbid necon1bbiddc
necon2abid necon2abiddc
necon2bbid necon2bbiddc
necon2abii necon2abiidc
necon2bbii necon2bbiidc
nebi nebidc
pm2.61ne, pm2.61ine, pm2.61dne, pm2.61dane, pm2.61da2ne, pm2.61da3ne pm2.61dc
neor pm2.53, ori, ord
neorian pm3.14
nnel none The reverse direction could be proved; the forward direction is double negation elimination.
nfrald nfraldxy, nfraldya
rexnal rexnalim
rexanali none
nrexralim none
dfral2 ralexim
dfrex2 rexalim, dfrex2dc
nfrexd nfrexdxy, nfrexdya
nfral nfralxy, nfralya
nfra2 nfra1, nfralya
nfrex nfrexxy, nfrexya
r19.30 none
r19.35 r19.35-1
ralcom2 ralcom
2reuswap 2reuswapdc
rmo2 rmo2i
df-pss and all proper subclass theorems none In set.mm, "A is a proper subclass of B" is defined to be (𝐴𝐵𝐴𝐵) and this definition is almost always used in conjunction with excluded middle. A more natural definition might be (𝐴𝐵 ∧ ∃𝑥𝑥 ∈ (𝐵𝐴)), if we need proper subclass at all.
nss nssr
ddif ddifnel, ddifss
df-symdif , dfsymdif2 symdifxor The symmetric difference notation and a number of the theorems could be brought over from set.mm.
dfss4 ssddif, dfss4st
dfun3 unssin
dfin3 inssun
dfin4 inssddif
unineq none
difindi difindiss
difdif2 difdif2ss
indm indmss
undif3 undif3ss
n0f n0rf
n0 n0r, notm0, fin0, fin0or
neq0 neq0r
reximdva0 reximdva0m
ssdif0 ssdif0im
inssdif0 inssdif0im
abn0 abn0r, abn0m
rabn0 rabn0m, rabn0r
csb0 csbconstg, csbprc The set.mm proof uses excluded middle to combine the 𝐴 ∈ V and ¬ 𝐴 ∈ V cases.
sbcel12 sbcel12g
sbcne12 sbcne12g
undif1 undif1ss
undif2 undif2ss
inundif inundifss
undif undifss subset rather than equality, for any sets
undiffi where both container and subset are finite
undifdc where the container has decidable equality and the subset is finite
undifdcss when membership in the subset is decidable
for any sets implies excluded middle as shown at undifexmid
forward direction only, for any sets still implies excluded middle as shown at exmidundifim
ssundif ssundifim
uneqdifeq uneqdifeqim
r19.2z r19.2m
r19.3rz r19.3rm
r19.28z r19.28m
r19.9rzv r19.9rmv
r19.37zv r19.3rmv
r19.45zv r19.45mv
r19.44zv r19.44mv
r19.27z r19.27m
r19.36zv r19.36av
dfif2 df-if
ifsb ifsbdc
dfif4 none Unused in set.mm
dfif5 none Unused in set.mm
ifnot none
ifor none
ifeq2da none
ifeqda none
elimif , ifval , elif , ifel , ifeqor , 2if2 , ifcomnan , csbif , csbifgOLD none
ifboth ifbothdc
ifid ifiddc
eqif eqifdc
ifan ifandc
dedth , dedth2h , dedth3h , dedth4h , dedth2v , dedth3v , dedth4v , elimhyp , elimhyp2v , elimhyp3v , elimhyp4v , elimel , elimdhyp , keephyp , keephyp2v , keephyp3v , keepel none Even in set.mm, the weak deduction theorem is discouraged in favor of theorems in deduction form.
ifpr none Should be provable if the condition is decidable.
difsnid difsnss One direction, for any set
nndifsnid for a natural number
dcdifsnid for a set with decidable equality
fidifsnid for a finite set
pwpw0 pwpw0ss that pwpw0 is equivalent to excluded middle follows from exmidpw or exmid01
sssn sssnr One direction, for any classes.
sssnm When the subset is inhabited.
for all sets Equivalent to excluded middle by exmidsssn.
ssunsn2 , ssunsn none
eqsn eqsnm
ssunpr none
sspr ssprr
sstp sstpr
prnebg none
pwsn pwsnss Also see exmidpw
pwtp pwtpss
pwpwpw0 pwpwpw0ss
iundif2 iundif2ss
iindif2 iindif2m
iinin2 iinin2m
iinin1 iinin1m
iinvdif iindif2m Unused in set.mm
riinn0 riinm
riinrab iinrabm
iunxdif3 none the set.mm proof relies on inundif
iinuni iinuniss
iununi iununir
rintn0 rintm
disjor disjnim
disjors disjnims
disji none
disjprg , disjxiun , disjxun none
disjss3 none Might need to be restated or have decidability conditions added
trintss trintssm
ax-rep ax-coll There are a lot of ways to state replacement and most/all of them hold, for example zfrep6 or funimaexg.
csbexg , csbex csbexga, csbexa set.mm uses case elimination to remove the 𝐴 ∈ V condition.
intex inteximm, intexr inteximm is the forward direction (but for inhabited rather than non-empty classes) and intexr is the reverse direction.
intexab intexabim
intexrab intexrabim
iinexg iinexgm Changes not empty to inhabited
intabs none Lightly used in set.mm, and the set.mm proof is not intuitionistic
reuxfr2d , reuxfr2 , reuxfrd , reuxfr none The set.mm proof of reuxfr2d relies on 2reuswap
moabex euabex In general, most of the set.mm ∃! theorems still hold, but a decent number of the ∃* ones get caught up on "there are two cases: the set exists or it does not"
snex snexg, snex

The iset.mm version of snex has an additional hypothesis.

We conjecture that the set.mm snex ({𝐴} ∈ V with no condition on whether 𝐴 is a set) is not provable.

The axioms of set theory allow us to construct rank levels from others, but not to construct something from nothing (except the 0 rank and stuff you can bootstrap from there). A class provides no "data" about where it lives, because it is spread out over the whole universe - you only get the ability to ask yes-no(-maybe) questions about set membership in the class. An assertion that a class exists is a piece of data that gives you a bound on the rank of this set, which can be used to build other existing things like unions and powersets of this class and so on. Anything with unbounded rank cannot be proven to exist.

So snex, which starts from an arbitrary class and produces evidence that { A } exists, cannot be provable, because the bound here can't depend on A and cannot be upper bounded by anything independent of A either - there are singletons at every rank.

However, we can refute a singleton being a proper class - see notnotsnex.

nssss nssssr
rmorabex euabex See discussion under moabex
nnullss mss
opex opexg, opex The iset.mm version of opex has additional hypotheses
otex otexg
opnz opm, opnzi
df-so df-iso Although we define Or to describe a weakly linear order (such as real numbers), there are some orders which are also trichotomous, for example nntri3or, pitri3or, and nqtri3or.
sotric sotricim One direction, for any weak linear order.
sotritric For a trichotomous order.
nntri2 For the specific order E Or ω
pitric For the specific order <N Or N
nqtric For the specific order <Q Or Q
sotrieq sotritrieq For a trichotomous order
sotrieq2 see sotrieq and then apply ioran
issoi issod, ispod Many of the set.mm usages of issoi don't carry over, so there is less need for this convenience theorem.
isso2i issod Presumably this could be proved if we need it.
df-fr df-frind
fri , dffr2 , frc none That any subset of the base set has an element which is minimal accordng to a well-founded relation presumably implies excluded middle (or is otherwise unprovable).
frss freq2 Because the definition of Fr is different than set.mm, the proof would need to be different.
frirr frirrg We do not yet have a lot of theorems for the case where 𝐴 is a proper class.
fr2nr none Shouldn't be hard to prove if we need it (using a proof similar to frirrg and en2lp).
frminex none Presumably unprovable.
efrn2lp none Should be easy but lightly used in set.mm
dfepfr , epfrc none Presumably unprovable.
df-we df-wetr
wess none See frss entry. Holds for E (see for example wessep).
weso wepo
wecmpep none We does not imply trichotomy in iset.mm
wefrc , wereu , wereu2 none Presumably not provable
dmxp dmxpm
relimasn imasng
xpnz xpm
xpeq0 xpeq0r, sqxpeq0
difxp none The set.mm proof relies on ianor
rnxp rnxpm
ssxpb ssxpbm
xp11 xp11m
xpcan xpcanm
xpcan2 xpcan2m
xpima2 xpima2m
sossfld , sofld , soex none the set.mm proofs rely on trichotomy
csbrn csbrng
dmsnn0 dmsnm
rnsnn0 rnsnm
relsn2 relsn2m
dmsnopss dmsnopg The domain is empty in the ¬ 𝐵 ∈ V case which follows readily from opprc2 and dmsn0. But we presumably cannot combine the 𝐵 ∈ V and ¬ 𝐵 ∈ V cases (set.mm uses excluded middle to do so).
dmsnsnsn dmsnsnsng
opswap opswapg
unixp unixpm
cnvso cnvsom
unixp0 unixp0im
unixpid none We could prove the theorem for the case where A is inhabited
cnviin cnviinm
xpco xpcom
xpcoid xpcom
tz7.7 none
ordelssne none
ordelpss none
ordsseleq , onsseleq onelss, eqimss, nnsseleq Taken together, onelss and eqimss represent the reverse direction of the biconditional from ordsseleq . For natural numbers the biconditional is provable.
ordtri3or nntri3or Ordinal trichotomy implies the law of the excluded middle as shown in ordtriexmid.
ordtri2 nntri2 ordtri2 for all ordinals presumably implies excluded middle although we don't have a specific proof analogous to ordtriexmid.
ordtri3 , ordtri4 , ordtri2or3 , dford2 none Ordinal trichotomy implies the law of the excluded middle as shown in ordtriexmid. We don't have similar proofs for every variation of of trichotomy but most of them are presumably powerful enough to imply excluded middle.
ordtri1 , ontri1 , onssneli , onssnel2i ssnel, nntri1 ssnel is a trichotomy-like theorem which does hold, although it is an implication whereas ordtri1 is a biconditional. nntri1 is biconditional, but just for natural numbers.
ordtr2 , ontr2 nntr2 ontr2 implies excluded middle as shown at ontr2exmid
ordtr3 none This is weak linearity of ordinals, which presumably implies excluded middle by ordsoexmid.
ord0eln0 , on0eln0 ne0i, nn0eln0
nsuceq0 nsuceq0g
ordsssuc trsucss, nnsssuc
ordequn none If you know which ordinal is larger, you can achieve a similar result via theorems such as oneluni or ssequn1.
ordun onun2
ordtri2or none Implies excluded middle as shown at ordtri2orexmid.
ordtri2or2 nntri2or2 ordtri2or2 implies excluded middle as shown at ordtri2or2exmid.
onsseli none See entry for ordsseleq
unizlim none The reverse direction is basically uni0 plus limuni
on0eqel 0elnn The full on0eqel is conjectured to imply excluded middle by an argument similar to ordtriexmid
snsn0non none Presumably would be provable (by first proving ¬ ∅ ∈ {{∅}} as in the set.mm proof, and then using that to show that {{∅}} is not a transitive set).
onxpdisj none Unused in set.mm
onnev none Presumably provable (see snsn0non entry)
nfiota nfiotaw
iotaex iotacl, euiotaex
dffun3 dffun5r
dffun5 dffun5r
resasplit resasplitss
fresaun none The set.mm proof relies on resasplit
fresaunres2 , fresaunres1 none The set.mm proof relies on resasplit
fint fintm
foconst none Although it presumably holds if non-empty is changed to inhabited, it would need a new proof and it is unused in set.mm.
f1oprswap none The 𝐴𝐵 case is handled (basically) by f1oprg. If there is a proof of the original f1oprswap which does not rely on case elimination, it would look very different from the set.mm proof.
dffv3 dffv3g
fvex funfvex when evaluating a function within its domain
fvexg, fvex when the function is a set and is evaluated at a set
relrnfvex when evaluating a relation whose range is a set
mptfvex when the function is defined via maps-to, yields a set for all inputs, and is evaluated at a set
1stexg, 2ndexg for the functions 1st and 2nd
slotex for a slot of an extensible structure
fvexi , fvexd see fvex
fvif fvifdc
fvrn0 none The set.mm proof uses case elimination on whether (𝐹𝑋) is the empty set.
fvssunirn fvssunirng, relfvssunirn
ndmfv ndmfvg The ¬ 𝐴 ∈ V case is fvprc.
elfvdm relelfvdm
elfvex , elfvexd relelfvdm, mptrcl
dffn5 dffn5im
fvmpti fvmptg The set.mm proof relies on case elimination on 𝐶 ∈ V
fvmpt2i fvmpt2 The set.mm proof relies on case elimination on 𝐶 ∈ V
fvmptss fvmptssdm
fvmptex none The set.mm proof relies on case elimination
fvmptnf none The set.mm proof relies on case elimination
fvmptn none The set.mm proof relies on case elimination
fvmptss2 fvmpt What fvmptss2 adds is the cases where this is a proper class, or we are out of the domain.
fvopab4ndm none
fndmdifeq0 none Although it seems like this might be intuitionizable, it is lightly used in set.mm.
f0cli ffvelrn
dff3 dff3im
dff4 dff4im
fmptsng , fmptsnd none presumably provable
fvunsn fvunsng
fnsnsplit fnsnsplitss Subset rather than equality, for a function on any set.
fnsnsplitdc For a function on a set with decidable equality.
fsnunf2 none Apparently would need decidable equality on 𝑆 or some other condition.
funresdfunsn funresdfunsnss Subset rather than equality, for a function on any set.
funresdfunsndc For a function on a set with decidable equality.
funiunfv fniunfv, funiunfvdm
funiunfvf funiunfvdmf
dff14a , dff14b none The set.mm proof depends, in an apparently essential way, on excluded middle.
fveqf1o none The set.mm proof relies on f1oprswap , which we don't have
soisores isoresbr
soisoi none The set.mm proof relies on trichotomy
isocnv3 none It seems possible that one direction of the biconditional could be proved.
isomin none
riotaex riotacl, riotaexg
csbriota , csbriotagOLD csbriotag
riotaxfrd none Although it may be intuitionizable, it is lightly used in set.mm. The set.mm proof relies on reuxfrd .
ovex fnovex when the operation is a function evaluated within its domain.
ovexg when the operation is a set and is evaluated at a set
relrnfvex when the operation is a relation whose range is a set
mpofvex When the operation is defined via maps-to, yields a set on any inputs, and is being evaluated at two sets.
addcl, expcl, etc If there is a closure theorem for a particular operation, that is often the way to intuitionize ovex (check for your particular operation, as these are just a few examples).
ovrcl elmpocl, relelfvdm
opabresex2d none it should be possible to update iset.mm to reflect set.mm in this area and related theorems
ovif12 none would be provable under the condition that 𝜑 is decidable
fnov fnovim
ov3 ovi3 Although set.mm's ov3 could be proved, it is only used a few places in set.mm, and in iset.mm those places need the modified form found in ovi3.
ndmovg , ndmov ndmfvg These theorems are generally used in set.mm for case elimination which is why we just have the general ndmfvg rather than an operation-specific version.
ndmovcl , ndmovcom , ndmovass , ndmovdistr , ndmovord , and ndmovordi none These theorems are generally used in set.mm for case elimination and the most straightforward way to avoid them is to add conditions that we are evaluating functions within their domains.
ndmovrcl elmpocl, relelfvdm
caov4 caov4d Note that caov4d has a closure hypothesis.
caov411 caov411d Note that caov411d has a closure hypothesis.
caov42 caov42d Note that caov42d has a closure hypothesis.
caovdir caovdird caovdird adds some constraints about where the operations are evaluated.
caovdilem caovdilemd
caovlem2 caovlem2d
caovmo caovimo
mpondm0 none could be proved, but usually is used in conjunction with excluded middle
elovmporab none could be proved, but unused in set.mm
elovmporab1 none could be proved
2mpo0 none Possibly provable, but an inhabited set version would be more likely to be helpful (if anything is).
relmptopab none Presumably would need a condition that 𝐵 ∈ dom 𝐹
ofval ofvalg
offn off the set.mm proof of offn uses ovex and it isn't clear whether anything can be proved with weaker hypotheses than off
offveq offeq
caofid0l , caofid0r , caofid1 , caofid2 none Assuming we really need to add conditions that the operations are functions being evaluated within their domains, there would be a fair bit of intuitionizing.
ordeleqon none
ssonprc none not provable (we conjecture), but interesting enough to intuitionize anyway. 𝐴 = On → 𝐴𝑉 is provable, and (𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴𝑉 is provable. (One thing we presumably could prove is ( 𝐴 ⊆ On ∧ ∃𝑥𝑥 ∈ (On ∖ 𝐴)) → 𝐴𝑉 which might be easier to understand if we define (or think of) proper subset as meaning that the set difference is inhabited.)
onint onintss onint implies excluded middle as shown in onintexmid.
onint0 none Thought to be "trivially not intuitionistic", and it is not clear if there is an alternate way to state it that is true. If the empty set is in A then of course |^| A = (/), but the converse seems difficult. I don't know so much about the structure of the ordinals without linearity,
onssmin, onminesb, onminsb none Conjectured to not be provable without excluded middle, for the same reason as onint.
oninton onintonm
onintrab none The set.mm proof relies on the converse of inteximm.
onintrab2 onintrab2im The converse would appear to need the converse of inteximm.
oneqmin none Falls as written because it implies onint, but it might be useful to keep the reverse direction for subsets that do have a minimum.
onminex none
onmindif2 none Conjectured to not be provable without excluded middle.
onmindif2 none Conjectured to not be provable without excluded middle.
ordpwsuc ordpwsucss See the ordpwsucss comment for discussion of the succcessor-like properites of (𝒫 𝐴 ∩ On). Full ordpwsuc implies excluded middle as seen at ordpwsucexmid.
ordsucelsuc onsucelsucr, nnsucelsuc The converse of onsucelsucr implies excluded middle, as shown at onsucelsucexmid.
ordsucsssuc onsucsssucr, nnsucsssuc The converse of onsucsssucr implies excluded middle, as shown at onsucsssucexmid.
ordsucuniel sucunielr, nnsucuniel Full ordsucuniel implies excluded middle, as shown at ordsucunielexmid.
ordsucun none yet Conjectured to be provable in the reverse direction, but not the forward direction (implies some order totality).
ordunpr none Presumably not provable without excluded middle.
ordunel none Conjectured to not be provable (ordunel implies ordsucun).
onsucuni, ordsucuni none Conjectured to not be provable without excluded middle.
orduniorsuc none Presumably not provable.
ordunisuc onunisuci, unisuc, unisucg
orduniss2 onuniss2
0elsuc none This theorem may appear to be innocuous but it implies excluded middle as shown at 0elsucexmid.
onuniorsuci none Conjectured to not be provable without excluded middle.
onuninsuci, orduninsuc none Conjectured to be provable in the forward direction but not the reverse one.
ordunisuc2 ordunisuc2r

The forward direction is conjectured to imply excluded middle. Here is a sketch of the proposed proof.

Let om' be the set of all finite iterations of suc' A = (𝒫 𝐴 ∩ On) on . (We can formalize this proof but not until we have om and at least finite induction.) Then om' = U. om' because if x e. om' then x = suc'^n (/) for some n, and then x C_ suc'^n (/) implies x e. suc'^(n+1) (/) e. om' so x e. U. om'.

Now supposing the theorem, we know that A. x e. om' suc x e. om', so in particular 2o e. om', that is, 2o = suc'^n (/) for some n. (Note that 1o = suc' (/) - see pw0.) For n = 0 and n = 1 this is clearly false, and for n = m+3 we have 1o e. suc' suc' (/) , so 2o C_ suc' suc' (/), so 2o e. suc' suc' suc' (/) C_ suc' suc' suc' suc'^m (/) = 2o, contradicting ordirr.

Thus 2o = suc' suc' (/) = suc' 1o. Applying this to X = {𝑥 ∈ {∅} ∣ 𝜑} we have X C_ 1o implies X e. suc' 1o = 2o and hence X = (/) \/ X = 1o, and LEM follows (by ordtriexmidlem2 for 𝑋 = ∅ and rabsnt as seen in the onsucsssucexmid proof for 𝑋 = 1o).

ordzsl, onzsl, dflim3, nlimon none
dflim4 df-ilim We conjecture that dflim4 is not equivalent to df-ilim.
limsuc none This would be trivial if dflim4 were the definition of a limit ordinal. With dflim2 as the definition, limsuc might need ordunisuc2 (which we believe is not provable, see its entry in this list).
limsssuc none Conjectured to be provable (is this also based on dflim4 being the definition of limit ordinal or is it unrelated?).
tfinds , tfindsg , tfindsg2 , tfindes , tfinds2 , tfinds3 tfis3 We are unable to separate limit and successor ordinals using case elimination.
df-om df-iom
findsg uzind4 findsg presumably could be proved, but there hasn't been a need for it.
xpexr none
xpexr2 xpexr2m
xpexcnv none would be provable if nonempty is changed to inhabited
1stval 1stvalg
2ndval 2ndvalg
1stnpr none May be intuitionizable, but very lightly used in set.mm.
2ndnpr none May be intuitionizable, but very lightly used in set.mm.
mptmpoopabbrd none it should be possible to update iset.mm to reflect set.mm in this area and related theorems
mptmpoopabovd none it should be possible to update iset.mm to reflect set.mm in this area and related theorems
el2mpocsbcl none the set.mm proof uses excluded middle
el2mpocl none the set.mm proof uses el2mpocsbcl
ovmptss none The set.mm proof relies on fvmptss
mposn none presumably provable
relmpoopab none The set.mm proof relies on ovmptss
mpoxeldm none presumably provable
mpoxneldm none presumably provable
mpoxopynvov0g none presumably provable
mpoxopxnop0 none presumably provable
mpoxopx0ov0 none presumably provable
mpoxopxprcov0 none presumably provable
mpoxopynvov0 none
mpoxopoveqd none unused in set.mm
brovmpoex none unused in set.mm
sprmpod none unused in set.mm
brtpos brtposg
ottpos ottposg
ovtpos ovtposg
df-cur , df-unc and all theorems using the curry and uncurry syntaxes none presumably could be added with only the usual issues around nonempty versus inhabited and the like
pwuninel pwuninel2 The set.mm proof of pwuninel uses case elimination.
iunonOLD iunon
smofvon2 smofvon2dm
tfr1 tfri1
tfr2 tfri2
tfr3 tfri3
tfr2b , recsfnon , recsval none These transfinite recursion theorems are lightly used in set.mm.
df-rdg df-irdg This definition combines the successor and limit cases (rather than specifying them as separate cases in a way which relies on excluded middle). In the words of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic", "we can still define many of the familiar set-theoretic operations by transfinite recursion on ordinals (see Aczel and Rathjen 2001, Section 4.2). This is fine as long as the definitions by transfinite recursion do not make case distinctions such as in the classical ordinal cases of successor and limit."
rdgfnon rdgifnon
ordge1n0 ordge1n0im, ordgt0ge1
ondif1 dif1o In set.mm, ondif1 is used for Cantor Normal Form
ondif2 , dif20el none The set.mm proof is not intuitionistic
brwitnlem none The set.mm proof is not intuitionistic
om0r om0, nnm0r
om00 nnm00
om00el none
omopth2 none The set.mm proof relies on ordinal trichotomy.
omeulem1 , omeu none The set.mm proof relies on ordinal trichotomy, omopth2 , oaord , and other theorems we don't have.
suc11reg suc11g
frfnom frecfnom frecfnom adopts the frec notation and adds conditions on the characteristic function and initial value.
fr0g frec0g frec0g adopts the frec notation and adds a condition on the characteristic function.
frsuc frecsuc frecsuc adopts the frec notation and adds conditions on the characteristic function and initial value.
om0x om0
oa0r none The set.mm proof distinguishes between limit and successor cases using case elimination.
oaordi nnaordi The set.mm proof of oaordi relies on being able to distinguish between limit ordinals and successor ordinals via case elimination.
oaord nnaord The set.mm proof of oaord relies on ordinal trichotomy.
oawordri none Implies excluded middle as shown at oawordriexmid
oaword oawordi The other direction presumably could be proven but isn't yet.
oaord1 none yet
oaword2 none The set.mm proof relies on oawordri and oa0r
oawordeu none The set.mm proof relies on a number of things we don't have
oawordex nnawordex The set.mm proof relies on oawordeu
omwordi nnmword The set.mm proof of omwordi relies on case elimination.
omword1 nnmword
oawordex nnawordex
oarec none the set.mm proof relies on tfinds3
oaf1o none the set.mm proof uses ontri1 and oawordeu
oacomf1o , oacomf1olem none the set.mm proof relies on oarec and oaf1o
swoso none Unused in set.mm.
ecdmn0 ecdmn0m
erdisj, qsdisj, qsdisj2, uniinqs none These could presumably be restated to be provable, but they are lightly used in set.mm
iiner iinerm
riiner riinerm
brecop2 none This is a form of reverse closure and uses excluded middle in its proof.
erov , erov2 none Unused in set.mm.
eceqoveq none Unused in set.mm.
ralxpmap none Lightly used in set.mm. The set.mm proof relies on fnsnsplit and undif .
nfixp nfixpxy, nfixp1 set.mm (indirectly) uses excluded middle to combine the cases where 𝑥 and 𝑦 are distinct and where they are not.
ixpexg ixpexgg
ixpiin ixpiinm
ixpint ixpintm
ixpn0 ixpm
undifixp none The set.mm proof relies on undif
resixpfo none The set.mm proof relies on membership in 𝐵 being decidable and would need to have nonempty changed to inhabited, but might be adaptable with those conditions added. However, this theorem is currently only used in the proof of Tychonoff's Theorem, which we do not expect to be able to prove.
boxriin none Would seem to need a condition that 𝐼 has decidable equality.
boxcutc none Would seem to need a condition that 𝐴 has decidable equality.
df-sdom , relsdom , brsdom , dfdom2 , sdomdom , sdomnen , brdom2 , bren2 , domdifsn none Many aspects of strict dominance as developed in set.mm rely on excluded middle and a different definition would be needed if we wanted strict dominance to have the expected properties.
en1b en1bg
snfi snfig
difsnen fidifsnen
undom none The set.mm proof uses undif2 and we just have undif2ss
xpdom3 xpdom3m
domunsncan none The set.mm proof relies on difsnen
omxpenlem , omxpen , omf1o none The set.mm proof relies on omwordi , oaord , om0r , and other theorems we do not have
pw2f1o , pw2eng , pw2en none Presumably would require some kind of decidability hypothesis. Discussions of this sort tend to get into how many truth values there are and sets like {𝑠𝑠 ⊆ 1o} (relatively undeveloped so far except for a few results like exmid01 and uni0b).
enfixsn none The set.mm proof relies on difsnen
sbth and its lemmas, sbthb , sbthcl fisbth The Schroeder-Bernstein Theorem is equivalent to excluded middle by exmidsbth
fodomr none Equivalent to excluded middle per exmidfodomr
mapdom1 mapdom1g
2pwuninel 2pwuninelg
mapunen none The set.mm proof relies on fresaunres1 and fresaunres2 .
map2xp none The set.mm proof relies on mapunen
mapdom2 , mapdom3 none The set.mm proof relies on case elimination and undif . At a minimum, it would appear that nonempty would need to be changed to inhabited.
pwen none The set.mm proof relies on pw2eng
limenpsi none The set.mm proof relies on sbth
limensuci , limensuc none The set.mm proof relies on undif
infensuc none The set.mm proof relies on limensuc
php phpm
snnen2o snnen2og, snnen2oprc
onomeneq , onfin none Not possible because they would imply onfin2
onfin2 none Implies excluded middle as shown as exmidonfin
nnsdomo , sucdom2 , sucdom , 0sdom1dom , sdom1 none iset.mm doesn't yet have strict dominance
1sdom2 1nen2 Although the presence of 1nen2 might make it look like a natural definition for strict dominance would be 𝐴𝐵 ∧ ¬ 𝐴𝐵, that definition may be more suitable for finite sets than all sets, so at least for now we only define and express certain theorems (such as this one) in terms of equinumerosity which in set.mm are expressed in terms of strict dominance.
modom , modom2 none The set.mm proofs rely on excluded middle
1sdom , unxpdom , unxpdom2 , sucxpdom none iset.mm doesn't yet have strict dominance
pssinf none The set.mm proof relies on sdomnen
isinf isinfinf
fineqv none The set.mm proof relies on theorems we don't have, and even for the theorems we do have, we'd need to carefully look at what axioms they rely on.
pssnn none The set.mm proof uses excluded middle.
ssnnfi none The proof in ssfiexmid would apply to this as well as to ssfi , since {∅} ∈ ω
ssfi ssfirab when the subset is defined by a decidable property
ssfidc when membership in the subset is decidable
for the general case Implies excluded middle as shown at ssfiexmid
domfi none Implies excluded middle as shown at domfiexmid
xpfir none Nonempty would need to be changed to inhabited, but the set.mm proof also uses domfi
infi none Implies excluded middle as shown at infiexmid. It is conjectured that we could prove the special case (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin
rabfi none Presumably the proof of ssfiexmid could be adapted to show this implies excluded middle
finresfin none The set.mm proof is in terms of ssfi
f1finf1o none The set.mm proof is not intuitionistic
nfielex none The set.mm proof relies on neq0
diffi diffisn, diffifi diffi is not provable, as shown at diffitest
enp1ilem , enp1i , en2 , en3 , en4 none The set.mm proof relies on excluded middle and undif1
findcard3 none The set.mm proof is in terms of strict dominance.
frfi none Not known whether this can be proved (either with the current df-frind or any other possible concept analogous to Fr).
fimax2g fimax2gtri
fimaxg none The set.mm proof of fimaxg relies on fimax2g which relies on frfi and fri
fisupg none The set.mm proof relies on excluded middle and presumably this theorem would need to be modified to be provable.
unbnn none the impossibility proof at exmidunben should apply here as well
unbnn2 none the impossibility proof at exmidunben should apply here as well
unfi unsnfi For the union of a set and a singleton whose element is not a member of that set
unfidisj For the union of two disjoint sets
unfiin When the intersection is known to be finite
for any two finite sets Implies excluded middle as shown at unfiexmid.
difinf difinfinf
prfi prfidisj for two unequal sets
in general The set.mm proof depends on unfi and it would appear that mapping {𝐴, 𝐵} to a natural number would decide whether 𝐴 and 𝐵 are equal and thus imply any set has decidable equality.
tpfi tpfidisj
fiint fiintim
fodomfi none Might be provable, for example via ac6sfi or induction directly. The set.mm proof does use undom in addition to induction.
fofinf1o none The set.mm proof uses excluded middle in several places.
dmfi fundmfi
resfnfinfin resfnfinfinss
residfi none Presumably provable, but lightly used in set.mm
cnvfi relcnvfi
rnfi funrnfi
fofi f1ofi Presumably precluded by an argument similar to domfiexmid (the set.mm proof relies on domfi).
iunfi iunfidisj for a disjoint collection
in general Presumably not possible for the same reasons as in unfiexmid
unifi none Presumably the issues are similar to iunfi
pwfi none The set.mm proof uses domfi and other theorems we don't have
abrexfi none At first glance it would appear that the mapping would need to be one to one or some other condition.
fisuppfi preimaf1ofi The set.mm proof of fisuppfi uses ssfi
intrnfi none presumably not provable; the set.mm proof uses fofi
iinfi none presumably not provable; the set.mm proof uses intrnfi
inelfi none may need a condition such as 𝐴𝐵; the set.mm proof uses prfi
fiin none presumably needs a condition analogous to those in unfidisj or unfiin; the set.mm proof uses unfi
dffi2 none the set.mm proof uses fiin and other theorems we do not have
inficl none the set.mm proof uses fiin and dffi2 (see also inelfi which might be relevant)
fipwuni none would need a 𝐴 ∈ V condition but even with that, the set.mm proof uses inficl
fisn none presumably would be provable (with an 𝐴 ∈ V condition added), but the set.mm proof uses inficl
fipwss fipwssg adds the condition that 𝐴 is a set
elfiun none the set.mm proof uses ssfi , fiin , and excluded middle
dffi3 none might be possible (perhaps using df-frec notation), but the set.mm proof does not work as-is
dfsup2 none The set.mm proof uses excluded middle in several places and the theorem is lightly used in set.mm.
supmo supmoti The conditions on the order are different.
supexd , supex none The set.mm proof uses rmorabex
supeu supeuti
supval2 supval2ti
eqsup eqsupti
eqsupd eqsuptid
supcl supclti
supub supubti
suplub suplubti
suplub2 suplub2ti
supnub none Presumably provable, although the set.mm proof relies on excluded middle and it is not used until later in set.mm.
sup0riota , sup0 , infempty none Suitably modified verions may be provable, but they are unused in set.mm.
supmax supmaxti
fisup2g , fisupcl none Other variations may be possible, but the set.mm proof will not work as-is or with small modifications.
supgtoreq none The set.mm proof uses fisup2g and also trichotomy.
suppr none The formulation using if would seem to require a trichotomous order. For real numbers, supremum on a pair does yield the maximum of two numbers: see maxcl, maxle1, maxle2, maxleast, and maxleb.
supiso supisoti
infexd , infex none See supexd
eqinf , eqinfd eqinfti, eqinftid
infval infvalti
infcllem cnvinfex infcllem has an unnecessary hypothesis; other than that these are the same
infcl infclti
inflb inflbti
infglb infglbti
infglbb none Presumably provable with additional conditions (see suplub2)
infnlb infnlbti
infmin infminti
infmo infmoti
infeu infeuti
fimin2g , fiming none The set.mm proof relies on frfi and fri
fiinfg , fiinf2g none The set.mm proof relies on fiming
fiinfcl none See fisupcl
infltoreq none The set.mm proof depends on supgtoreq and fiinfcl
infpr none See suppr
infsn infsnti
infiso infisoti
ax-reg , axreg2 , zfregcl ax-setind ax-reg implies excluded middle as seen at regexmid
ax-inf , zfinf , axinf2 , axinf ax-iinf, zfinf2 probably most of the variations of the axiom of infinity in set.mm could be shown to be equivalent to each other in iset.mm as well
inf5 none as this is in terms of proper subset it is unlikely to work well in iset.mm
elom3 , dfom4 , dfom5 none the set.mm theorems depend on theorems we do not have
oancom none presumably provable but the set.mm proof does not work as-is
isfinite fict
omenps , omensuc none presumably provable but the set.mm proofs do not work as-is
infdifsn none possibly provable with added 𝑥𝐴𝑦𝐴DECID 𝑥 = 𝑦 and 𝐵𝐴 conditions, for example via a technique analogous to difinfsn
infdiffi none if we can prove a version of infdifsn it should carry over from singletons to finite sets, much the way difinfinf follows from difinfsn
unbnn3 none the impossibility proof at exmidunben should apply here as well
noinfep none the set.mm proof uses fri
df-rank and all theorems related to the rank function none One possible definition is Definition 9.3.4 of [AczelRathjen], p. 91
df-aleph and all theorems involving aleph none
df-cf and all theorems involving cofinality none
df-acn and all theorems using this definition none
cardf2 , cardon , isnum2 cardcl, isnumi It would also be easy to prove Fun card if there is a need.
ennum none The set.mm proof relies on isnum2
tskwe none Relies on df-sdom
xpnum none The set.mm proof relies on isnum2
cardval3 cardval3ex
cardid2 none The set.mm proof relies on onint
isnum3 none
oncardid none The set.mm proof relies on cardid2
cardidm none Presumably this would need a condition on 𝐴 but even with that, the set.mm proof relies on cardid2
oncard none The set.mm proof relies on theorems we don't have, and this theorem is unused in set.mm.
ficardom none Presumably not possible for the same reasons as onfin2
ficardid none The set.mm proof relies on cardid2
cardnn none The set.mm proof relies on a variety of theorems we don't have currently.
cardnueq0 none The set.mm proof relies on cardid2
cardne none The set.mm proof relies on ordinal trichotomy (and if that can be solved there might be some more minor problems which require revisions to the theorem)
carden2a none The set.mm proof relies on excluded middle.
carden2b carden2bex
card1 , cardsn none Rely on a variety of theorems we don't currently have. Lightly used in set.mm.
carddomi2 none The set.mm proof relies on excluded middle.
sdomsdomcardi none Relies on a variety of theorems we don't currently have.
prdom2 none The set.mm proof would only work in the case where 𝐴 = 𝐵 is decidable. If we can prove fodomfi , that would appear to imply prsomd2 fairly quickly.
dif1card none The set.mm proof relies on cardennn
leweon none We lack the well ordering related theorems this relies on, and it isn't clear they are provable.
r0weon none We lack the well ordering related theorems this relies on, and it isn't clear they are provable.
infxpen none The set.mm proof relies on well ordering related theorems that we don't have (and may not be able to have), and it isn't clear that infxpen is provable.
infxpidm2 none Depends on cardinality theorems we don't have.
infxpenc none Relies on notations and theorems we don't have.
infxpenc2 none Relies on theorems we don't have.
dfac8a none The set.mm proof does not work as-is and numerability may not be able to work the same way.
dfac8b none We are lacking much of what this proof relies on and we may not be able to make numerability and well-ordering work as in set.mm.
dfac8c none The set.mm proof does not work as-is and well-ordering may not be able to work the same way.
ac10ct none The set.mm proof does not work as-is and well-ordering may not be able to work the same way.
ween none The set.mm proof does not work as-is and well-ordering and numerability may not be able to work the same way.
ac5num , ondomen , numdom , ssnum , onssnum , indcardi none Numerability (or cardinality in general) is not well developed and to a certain extent cannot be.
undjudom none The set.mm proof relies on undom
dju1dif none presumably provable
mapdjuen none the set.mm proof relies on mapunen
pwdjuen none the set.mm proof relies on mapdjuen
djudom1 none the set.mm proof relies on undom
djudom2 none the set.mm proof relies on djudom1
djuxpdom none presumably provable if having cardinality greater than one is expressed as 2o𝐴 instead
djufi none we should be able to prove (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin
cdainflem , djuinf none the set.mm proof is not easily adapted
infdju1 none the set.mm proof relies on infdifsn
pwdju1 none the set.mm proof relies on pwdjuen and pwpw0
pwdjuidm none the set.mm proof relies on infdju1 and pwdju1
djulepw none
onadju none the set.mm proof uses oacomf1olem and oarec
cardadju none the set.mm proof uses cardon , onadju , and cardid2
djunum none the set.mm proof uses cardon and cardadju
unnum none the set.mm proof uses djunum , numdom , and undjudom
nnadju none depends on various cardinality theorems we don't have
ficardun none depends on various cardinality theorems we don't have
ficardun2 none depends on various cardinality theorems we don't have
pwsdompw none depends on various cardinality theorems we don't have
unctb unct
infdjuabs none the set.mm proof uses sbth and other theorems we don't have
infunabs none the set.mm proof uses undjudom and infdjuabs
infdju none the set.mm proof uses sbth and other theorems we don't have
infdif none the set.mm proof uses sbth and other theorems we don't have
pwdjudom none
ax-cc cc1
axcc2 cc2
axcc3 cc3
axcc4 cc4
fodom , fodomnum none Presumably not provable as stated
fodomb none That the reverse direction is equivalent to excluded middle is exmidfodomr. The forward direction is presumably also not provable.
entri3 fientri3 Because full entri3 is equivalent to the axiom of choice, it implies excluded middle.
infinf infnfi Defining "A is infinite" as ω ≼ 𝐴 follows definition 8.1.4 of [AczelRathjen], p. 71. It can presumably not be shown to be equivalent to ¬ 𝐴 ∈ Fin in the absence of excluded middle (see inffiexmid which isn't exactly about ¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴 but which is close).
df-wina , df-ina , df-tsk , df-gru , ax-groth and all theorems related to inaccessibles and large cardinals none For an introduction to inaccessibles and large set properties see Chapter 18 of [AczelRathjen], p. 165 (including why "large set properties" is more apt terminology than "large cardinal properties" in the absence of excluded middle).
df-wun and all weak universe theorems none
mulcompi mulcompig
mulasspi mulasspig
distrpi distrpig
mulcanpi mulcanpig
ltapi ltapig
ltmpi ltmpig
nlt1pi nlt1pig
df-nq df-nqqs
df-nq df-nqqs
df-erq none Not needed given df-nqqs
df-plq df-plqqs
df-mq df-mqqs
df-1nq df-1nqqs
df-ltnq df-ltnqqs
elpqn none Not needed given df-nqqs
ordpipq ordpipqqs
addnqf dmaddpq, addclnq It should be possible to prove that +Q is a function, but so far there hasn't been a need to do so.
mulcomnq mulcomnqg
mulassnq mulassnqg
recmulnq recmulnqg
ltanq ltanqg
ltmnq ltmnqg
ltexnq ltexnqq
archnq archnqq
df-np df-inp
df-1p df-i1p
df-plp df-iplp
df-ltp df-iltp
elnp , elnpi elinp
prn0 prml, prmu
elprnq elprnql, elprnqu
prcdnq prcdnql, prcunqu
prub prubl
prnmax prnmaxl
npomex none
genpv genipv
genpcd genpcdl
genpnmax genprndl
ltrnq ltrnqg, ltrnqi
genpass genpassg
plpv plpvlu
mpv mpvlu
nqpr nqprlu
mulclprlem mulnqprl, mulnqpru
mulcompr mulcomprg
mulasspr mulassprg
distrlem1pr distrlem1prl, distrlem1pru
distrlem4pr distrlem4prl, distrlem4pru
distrlem5pr distrlem5prl, distrlem5pru
distrpr distrprg
ltprord ltprordil There hasn't yet been a need to investigate versions which are biconditional or which involve proper subsets.
psslinpr ltsopr
prlem934 prarloc2
ltexprlem1 , ltexprlem2 , ltexprlem3 , ltexprlem4 none See the lemmas for ltexpri generally.
ltexprlem5 ltexprlempr
ltexprlem6 ltexprlemfl, ltexprlemfu
ltexprlem7 ltexprlemrl, ltexprlemru
ltapr ltaprg
prlem936 prmuloc2
reclem2pr recexprlempr
reclem3pr recexprlem1ssl, recexprlem1ssu
reclem4pr recexprlemss1l, recexprlemss1u, recexprlemex
supexpr suplocexpr also see caucvgprpr but completeness cannot be formulated as in set.mm without changes
mulcmpblnrlem mulcmpblnrlemg
ltsrpr ltsrprg
dmaddsr , dmmulsr none Although these presumably could be proved in a way similar to dmaddpq and dmmulpq (in fact dmaddpqlem would seem to be easily generalizable to anything of the form ((𝑆 × 𝑇) / 𝑅)), we haven't yet had a need to do so.
mulcomsr mulcomsrg
mulasssr mulasssrg
distrsr distrsrg
ltasr ltasrg
sqgt0sr mulgt0sr, apsqgt0
recexsr recexsrlem This would follow from sqgt0sr (as in the set.mm proof of recexsr), but "not equal to zero" would need to be changed to "apart from zero".
mappsrpr mappsrprg
ltpsrpr ltpsrprg
map2psrpr map2psrprg
supsr suplocsr also see caucvgsr
ax1ne0 , ax-1ne0 ax0lt1, ax-0lt1, 1ap0, 1ne0
axrrecex , ax-rrecex axprecex, ax-precex
axpre-lttri , ax-pre-lttri axpre-ltirr, axpre-ltwlin, ax-pre-ltirr, ax-pre-ltwlin
axpre-sup , ax-pre-sup axpre-suploc, ax-pre-suploc
axsup axsuploc
elimne0 none Even in set.mm, the weak deduction theorem is discouraged in favor of theorems in deduction form.
xrltnle xrlenlt
ssxr df-xr Lightly used in set.mm
ltnle , ltnlei , ltnled lenlt, zltnle
lttri2 , lttri2i , lttri2d qlttri2 Real number trichotomy is not provable.
lttri4 ztri3or, qtri3or Real number trichotomy is not provable.
leloe , eqlelt , leloei , leloed , eqleltd none
leltne , leltned leltap, leltapd
ltneOLD ltne, ltap
letric , letrii , letrid zletric, qletric
ltlen , ltleni , ltlend ltleap, zltlen, qltlen
ne0gt0 , ne0gt0d ap0gt0, ap0gt0d
lecasei , ltlecasei none These are real number trichotomy
lelttric zlelttric, qlelttric
lttrid , lttri4d none These are real number trichotomy
leneltd leltapd
dedekind , dedekindle dedekindeu various details are different including that in dedekindeu the lower and upper cuts have to be open (and thus the real corresponding to the Dedekind cut is not contained in either the lower or upper cut)
mul02lem1 none The one use in set.mm is not needed in iset.mm.
negex negcl
msqgt0 , msqgt0i , msqgt0d apsqgt0 "Not equal to zero" is changed to "apart from zero"
relin01 none Relies on real number trichotomy.
ltord1 , leord1 , ltord2 , leord2 none The set.mm proof relies on real number trichotomy.
wloglei , wlogle none These depend on real number trichotomy and are not used until later in set.mm.
recex recexap In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mulcand, mulcan2d mulcanapd, mulcanap2d In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mulcan , mulcan2 , mulcani mulcanap, mulcanap2, mulcanapi In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mul0or , mul0ori , mul0ord mul0eqap Remark 2.19 of [Geuvers] says that (𝐴 · 𝐵) = 0 → (𝐴 = 0 ∨ 𝐵 = 0) does not hold in general and has a counterexample.
mulne0 , mulne0i , mulne0d mulap0, mulap0i, mulap0d
receu receuap In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mulnzcnopr none
msq0i , msq0d sqeq0, sqeq0i These slight restatements of sqeq0 are unused in set.mm.
mulcan1g , mulcan2g various cancellation theorems Presumably this is unavailable for the same reason that mul0or is unavailable.
1div0 none This could be proved, but the set.mm proof does not work as-is.
divval divvalap In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divmul , divmul2 , divmul3 divmulap, divmulap2, divmulap3 In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divcl , reccl divclap, recclap In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divcan1 , divcan2 divcanap1, divcanap2 In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
diveq0 diveqap0 In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divne0b , divne0 divap0b, divap0
recne0 recap0
recid , recid2 recidap, recidap2
divrec divrecap
divrec2 divrecap2
divass divassap
div23 , div32 , div13 , div12 div23ap, div32ap, div13ap, div12ap
divmulass divmulassap
divmulasscom divmulasscomap
divdir , divcan3 , divcan4 divdirap, divcanap3, divcanap4
div11 , divid , div0 div11ap, dividap, div0ap
diveq1 , divneg diveqap1, divnegap
muldivdir muldivdirap
divsubdir divsubdirap
recrec , rec11 , rec11r recrecap, rec11ap, rec11rap
divmuldiv , divdivdiv , divcan5 divmuldivap, divdivdivap, divcanap5
divmul13 , divmul24 , divmuleq divmul13ap, divmul24ap, divmuleqap
recdiv , divcan6 , divdiv32 , divcan7 recdivap, divcanap6, divdiv32ap, divcanap7
dmdcan , divdiv1 , divdiv2 , recdiv2 dmdcanap, divdivap1, divdivap2, recdivap2
conjmul , rereccl, redivcl conjmulap, rerecclap, redivclap
div2neg , divneg2 div2negap, divneg2ap
recclzi , recne0zi , recidzi recclapzi, recap0apzi, recidapzi
reccli , recidi , recreci recclapi, recidapi, recrecapi
dividi , div0i dividapi, div0api
divclzi , divcan1zi , divcan2zi divclapzi, divcanap1zi, divcanap2zi
divreczi , divcan3zi , divcan4zi divrecapzi, divcanap3zi, divcanap4zi
rec11i , rec11ii rec11api, rec11apii
divcli , divcan2i , divcan1i , divreci , divcan3i , divcan4i divclapi, divcanap2i, divcanap1i, divrecapi, divcanap3i, divcanap4i
div0i divap0i
divasszi , divmulzi , divdirzi , divdiv23zi divassapzi, divmulapzi, divdirapzi, divdiv23apzi
divmuli , divdiv32i divmulapi, divdiv32api
divassi , divdiri , div23i , div11i divassapi, divdirapi, div23api, div11api
rerecclzi , rereccli , redivclzi , redivcli rerecclapzi, rerecclapi, redivclapzi, redivclapi
reccld , rec0d , recidd , recid2d , recrecd , dividd , div0d recclapd, recap0d, recidapd, recidap2d, recrecapd, dividapd, div0apd
divcld , divcan1d , divcan2d , divrecd , divrec2d , divcan3d , divcan4d divclapd, divcanap1d, divcanap2d, divrecapd, divrecap2d, divcanap3d, divcanap4d
diveq0d , diveq1d , diveq1ad , diveq0ad , divne1d , div0bd , divnegd , divneg2d , div2negd diveqap0d, diveqap1d, diveqap1ad, diveqap0ad, divap1d, divap0bd, divnegapd, divneg2apd, div2negapd
divne0d , recdivd , recdiv2d , divcan6d , ddcand , rec11d divap0d, recdivapd, recdivap2d, divcanap6d, ddcanapd, rec11apd
divmuld , div32d , div13d , divdiv32d , divcan5d , divcan5rd , divcan7d , dmdcand , dmdcan2d , divdiv1d , divdiv2d divmulapd, div32apd, div13apd, divdiv32apd, divcanap5d, divcanap5rd, divcanap7d, dmdcanapd, dmdcanap2d, divdivap1d, divdivap2d
divmul2d, divmul3d, divassd, div12d, div23d, divdird, divsubdird, div11d divmulap2d, divmulap3d, divassapd, div12apd, div23apd, divdirapd, divsubdirapd, div11apd
divmuldivd divmuldivapd
rereccld , redivcld rerecclapd, redivclapd
diveq1bd diveqap1bd
div2sub , div2subd div2subap, div2subapd
subrec , subreci , subrecd subrecap, subrecapi, subrecapd
mvllmuld mvllmulapd
mvllmuli mvllmulapd
elimgt0 , elimge0 none Even in set.mm, the weak deduction theorem is discouraged in favor of theorems in deduction form.
mulge0b , mulsuble0b none Presumably unprovable for reasons analogous to mul0or.
mulle0b mulle0r The converse of mulle0r is presumably unprovable for reasons analogous to mul0or.
ledivp1i , ltdivp1i none Presumably could be proved, but unused in set.mm.
fimaxre fimaxq, fimaxre2 When applied to a pair fimaxre could show which of two unequal real numbers is larger, so perhaps not provable for that reason. (see fin0 for inhabited versus nonempty).
fimaxre3 none The set.mm proof relies on abrexfi .
fiminre none See fimaxre
sup2 none Presumably provable (with a locatedness condition added) but most likely not as interesting as sup3 in the absence of leloe
sup3 , sup3ii none We won't be able to have the least upper bound property for all inhabited bounded sets, as shown at sup3exmid.
infm3 none See sup3
suprcl , suprcld , suprclii supclti
suprub , suprubd , suprubii suprubex
suprlub , suprlubii suprlubex
suprnub , suprnubii suprnubex
suprleub , suprleubii suprleubex
supaddc , supadd none Presumably provable with suitable conditions for the existence of the supremums
supmul1 , supmul none Presumably provable with suitable conditions for the existence of the supremums
riotaneg none The theorem is unused in set.mm and the set.mm proof relies on reuxfrd
infrecl infclti
infrenegsup infrenegsupex
infregelb none yet Presumably could be handled in a way analogous to suprleubex
infrelb none yet Presumably could be handled in a way analogous to suprubex
supfirege suprubex The question here is whether results like maxle1 can be generalized (presumably by induction) from pairs to finite sets.
crne0 crap0
ofsubeq0 , ofnegsub , ofsubge0 none Depend on ofval and/or offn .
df-nn dfnn2
dfnn3 dfnn2 Presumably could be proved, as it is a slight variation of dfnn2
avgle qavgle
nnunb none Presumably provable from arch but unused in set.mm.
frnnn0supp , frnnn0fsupp nn0supp iset.mm does not yet have either the notation, or in some cases the theorems, related to the support of a function or a fintely supported function.
suprzcl suprzclex
zriotaneg none Lightly used in set.mm
suprfinzcl none
decex deccl
uzwo , uzwo2 , nnwo , nnwof , nnwos none Presumably would imply excluded middle, unless there is something which makes this different from nnregexmid.
negn0 negm
uzinfi, nninf, nn0inf none Presumably provable
infssuzle infssuzledc
infssuzcl infssuzcldc
supminf supminfex
zsupss , suprzcl2 zsupcl, suprzclex
suprzub none Presumably could prove something like this with different conditions for the existence of the supremum (see infssuzledc for something along these lines).
uzsupss zsupcl
uzwo3 , zmin none Proved in terms of supremum theorems and presumably not possible without excluded middle.
zbtwnre none Proved in terms of supremum theorems and presumably not possible without excluded middle.
rebtwnz qbtwnz
rpneg rpnegap
mul2lt0bi mul2lt0np, mul2lt0pn The set.mm proof of the forward direction of mul2lt0bi is not intuitionistic.
xrlttri , xrlttri2 none A generalization of real trichotomy.
xrleloe , xrleltne , dfle2 none Consequences of real trichotomy.
xrltlen none We presumably could prove an analogue to ltleap but we have not yet defined apartness for extended reals (# is for complex numbers).
dflt2 none
xrletri none
xrmax1 xrmax1sup
xrmax2 xrmax2sup
xrmin1 , xrmin2 xrmin1inf, xrmin2inf
xrmaxeq xrmaxleim
xrmineq xrmineqinf
xrmaxlt xrmaxltsup
xrltmin xrltmininf
xrmaxle xrmaxlesup
xrlemin xrlemininf
max0sub , ifle none
max1 maxle1
max2 maxle2
2resupmax 2zsupmax for integers
in general The set.mm proof uses real trichotomy in an apparently essential way. We express maximum in iset.mm using sup({𝐴, 𝐵}, ℝ, < ) rather than if(𝐴𝐵, 𝐵, 𝐴). The former has the expected maximum properties such as maxcl, maxle1, maxle2, maxleast, and maxleb.
ssfzunsnext none Presumably provable using 2zsupmax and similar theorems.
min1 min1inf
min2 min2inf
maxle maxleastb
lemin lemininf
maxlt maxltsup
ltmin ltmininf
lemaxle maxle2
qsqueeze none yet Presumably provable from qbtwnre and squeeze0, but unused in set.mm.
qextltlem , qextlt , qextle none The set.mm proof is not intuitionistic.
xralrple , alrple none yet Now that we have qbtwnxr, it looks like the set.mm proof would work with minor changes.
xnegex xnegcl
xmulval none The set.mm definition would appear to only function if comparing a real number with zero is decidable (which we will not be able to show in general)
xmullem , xmullem2 , xmulcom , xmul01 , xmul02 , xmulneg1 , xmulneg2 , rexmul , xmulf , xmulcl , xmulpnf1 , xmulpnf2 , xmulmnf1 , xmulmnf2 , xmulpnf1n , xmulid1 , xmulid2 , xmulm1 , xmulasslem2 , xmulgt0 , xmulge0 , xmulasslem , xmulasslem3 , xmulass , xlemul1a , xlemul2a , xlemul1 , xlemul2 , xltmul1 , xltmul2 , xadddilem , xadddi , xadddir , xadddi2 , xadddi2r , x2times , xmulcld none Although a few of these contain hypotheses that arguments are apart from zero (and thus could be proved with the current definition of ·e), in general extended real multiplication will not work as in set.mm using that definition.
xrsupss none the set.mm proof relies on sup3 which implies excluded middle by sup3exmid
supxrcl none the set.mm proof relies on xrsupss
infxrcl none presumably not provable for all the usual sup3exmid reasons
ixxub , ixxlb none
supicc , supiccub , supicclub , supicclub2 suplociccreex, suplociccex
ixxun , ixxin none
ioon0 ioom Non-empty is changed to inhabited
iooid iooidg
ndmioo none See discussion at ndmov but set.mm uses excluded middle, both in proving this and in using it.
lbioo , ubioo lbioog, ubioog
iooin iooinsup
icc0 icc0r
ioorebas ioorebasg
ge0xmulcl none Relies on xmulcl ; see discussion in this list for that theorems.
icoun , snunioo , snunico , snunioc , prunioo none
ioojoin none
difreicc none
iccsplit none This depends, apparently in an essential way, on real number trichotomy.
xov1plusxeqvd none This presumably could be proved if not equal is changed to apart, but is lightly used in set.mm.
fzn0 fzm
fz0 none Although it would be possible to prove a version of this with the additional conditions that 𝑀 ∈ V and 𝑁 ∈ V, the theorem is lightly used in set.mm.
fzon0 fzom
fzo0n0 fzo0m
ssfzoulel none Presumably could be proven, but the set.mm proof is not intuitionistic and it is lightly used in set.mm.
fzonfzoufzol none Presumably could be proven, but the set.mm proof is not intuitionistic and it is lightly used in set.mm.
elfznelfzo , elfznelfzob , injresinjlem , injresinj none Some or all of this presumably could be proven, but the set.mm proof is not intuitionistic and it is lightly used in set.mm.
flcl , reflcl , flcld flqcl, flqcld
fllelt flqlelt
flle flqle
flltp1 , fllep1 flqltp1
fraclt1 , fracle1 qfraclt1
fracge0 qfracge0
flge flqge
fllt flqlt
flflp1 none The set.mm proof relies on case elimination.
flidm flqidm
flidz flqidz
flltnz flqltnz
flwordi flqwordi
flword2 flqword2
flval2 , flval3 none Unused in set.mm
flbi flqbi
flbi2 flqbi2
ico01fl0 none Presumably could be proved for rationals, but lightly used in set.mm.
flge0nn0 flqge0nn0
flge1nn flqge1nn
refldivcl flqcl
flmulnn0 flqmulnn0
fldivle flqle
ltdifltdiv none Unused in set.mm.
fldiv4lem1div2uz2 , fldiv4lem1div2 none Presumably provable, but lightly used in set.mm.
ceilval ceilqval The set.mm ceilval, with a real argument and no additional conditions, is probably provable if there is a need.
dfceil2 , ceilval2 none Unused in set.mm.
ceicl ceiqcl
ceilcl ceilqcl
ceilge ceilqge
ceige ceiqge
ceim1l ceiqm1l
ceilm1lt ceilqm1lt
ceile ceiqle
ceille ceilqle
ceilidz ceilqidz
flleceil flqleceil
fleqceilz flqeqceilz
quoremz , quoremnn0 , quoremnn0ALT none Unused in set.mm.
intfrac2 intqfrac2
fldiv flqdiv
fldiv2 none Presumably would be provable if real is changed to rational.
fznnfl none Presumably would be provable if real is changed to rational.
uzsup , ioopnfsup , icopnfsup , rpsup , resup , xrsup none As with most theorems involving supremums, these would likely need significant changes
modval modqval As with theorems such as flqcl, we prove most of the modulo related theorems for rationals, although other conditions on real arguments other than whether they are rational would be possible in the future.
modvalr modqvalr
modcl , modcld modqcl, modqcld
flpmodeq flqpmodeq
mod0 modq0
mulmod0 mulqmod0
negmod0 negqmod0
modge0 modqge0
modlt modqlt
modelico modqelico
moddiffl modqdiffl
moddifz modqdifz
modfrac modqfrac
flmod flqmod
intfrac intqfrac
modmulnn modqmulnn
modvalp1 modqvalp1
modid modqid
modid0 modqid0
modid2 modqid2
0mod q0mod
1mod q1mod
modabs modqabs
modabs2 modqabs2
modcyc modqcyc
modcyc2 modqcyc2
negmod qnegmod
modm1p1mod0 modqm1p1mod0
modltm1p1mod modqltm1p1mod
modmul1 modqmul1
modmul12d modqmul12d
modnegd modqnegd
modsub12d modqsub12d
modsubmod modqsubmod
modsubmodmod modqsubmodmod
2txmodxeq0 q2txmodxeq0
2submod q2submod
modmulmod modqmulmod
modmulmodr modqmulmodr
moddi modqdi
modsubdir modqsubdir
modeqmodmin modqeqmodmin
modirr none A version of this (presumably modified) may be possible, but it is unused in set.mm
om2uz0i frec2uz0d
om2uzsuci frec2uzsucd
om2uzuzi frec2uzuzd
om2uzlti frec2uzltd
om2uzlt2i frec2uzlt2d
om2uzrani frec2uzrand
om2uzf1oi frec2uzf1od
om2uzisoi frec2uzisod
om2uzoi , ltweuz , ltwenn , ltwefz none Based on theorems like nnregexmid it is not clear what, if anything, along these lines is possible.
om2uzrdg frec2uzrdg
uzrdglem frecuzrdglem
uzrdgfni frecuzrdgtcl
uzrdg0i frecuzrdg0
uzrdgsuci frecuzrdgsuc
uzinf none See ominf
uzrdgxfr none Presumably could be proved if restated in terms of frec (a la frec2uz0d). However, it is lightly used in set.mm.
fzennn frecfzennn
fzen2 frecfzen2
cardfz none Cardinality does not work the same way without excluded middle and iset.mm has few cardinality related theorems.
hashgf1o frechashgf1o
fzfi fzfig
fzfid fzfigd
fzofi fzofig
fsequb none Seems like it might be provable, but unused in set.mm
fsequb2 none The set.mm proof does not work as-is
fseqsupcl none The set.mm proof relies on fisupcl and it is not clear whether this supremum theorem or anything similar can be proved.
fseqsupubi none The set.mm proof relies on fsequb2 and suprub and it is not clear whether this supremum theorem or anything similar can be proved.
uzindi none This could presumably be proved, perhaps from uzsinds, but is lightly used in set.mm
axdc4uz none Although some versions of constructive mathematics accept dependent choice, we have not yet developed it in iset.mm
ssnn0fi , rabssnn0fi none Conjectured to imply excluded middle along the lines of nnregexmid or ssfiexmid
df-seq df-seqfrec
seqval seq3val
seqfn seqf
seq1 , seq1i seq3-1
seqp1 , seqp1i seq3p1
seqm1 seq3m1
seqcl2 seqf2 seqf2 requires that 𝐹 be defined on (ℤ𝑀) not merely (𝑀...𝑁).
seqcl seqf, seq3clss seqf requires that 𝐹 be defined on (ℤ𝑀) not merely (𝑀...𝑁). This requirement is relaxed somewhat in seq3clss.
seqfveq2 seq3fveq2
seqfeq2 seq3feq2
seqfveq seq3fveq
seqfeq seq3feq
seqshft2 seq3shft2
seqres none Should be intuitionizable as with the other seq theorems, but unused in set.mm
sermono ser3mono ser3mono requires that 𝐹 be defined on (ℤ𝑀) not merely (𝑀...𝑁) as in sermono .
seqsplit seq3split seq3split requires that 𝐹 be defined on (ℤ𝐾) not merely (𝐾...𝑁) as in seqsplit . This is not a problem when used on infinite sequences; finite sums may find it easier to use fsumsplit instead.
seq1p seq3-1p Requires that 𝐹 be defined on (ℤ𝑀) not merely (𝑀...𝑁). This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqcaopr3 seq3caopr3 The functions 𝐹, 𝐺, and 𝐻 need to be defined on (ℤ𝑀) not merely (𝑀...𝑁).
seqcaopr2 seq3caopr2 The functions 𝐹, 𝐺, and 𝐻 need to be defined on (ℤ𝑀) not merely (𝑀...𝑁).
seqcaopr seq3caopr The functions 𝐹, 𝐺, and 𝐻 need to be defined on (ℤ𝑀) not merely (𝑀...𝑁).
seqf1o seq3f1o The functions 𝐺 and 𝐻 need to be defined on (ℤ𝑀) not merely (𝑀...𝑁). Also, a single set 𝑆 takes the place of 𝐶 and 𝑆 because that is sufficient flexibility at least for now.
seradd ser3add The functions 𝐹, 𝐺, and 𝐻 need to be defined on (ℤ𝑀) not merely (𝑀...𝑁).
sersub ser3sub The functions 𝐹, 𝐺, and 𝐻 need to be defined on (ℤ𝑀) not merely (𝑀...𝑁).
seqid3 seq3id3
seqid seq3id
seqid2 seq3id2
seqhomo seq3homo
seqz seq3z The sequence has to be defined on (ℤ𝑀) not just (𝑀...𝑁)
seqfeq4 seqfeq3 The sequence has to be defined on (ℤ𝑀) not just (𝑀...𝑁)
seqdistr seq3distr
serge0 ser3ge0 The sequence has to be defined on (ℤ𝑀) not just (𝑀...𝑁)
serle ser3le Changes several hypotheses from (𝑀...𝑁) to (ℤ𝑀)
ser1const fsumconst Finite summation in iset.mm is easier to express using Σ rather than seq directly.
seqof , seqof2 none It should be possible to come up with some (presumably modified) versions of these, but we have not done so yet.
expval exp3val The set.mm theorem does not exclude the case of dividing by zero.
expneg expnegap0 The set.mm theorem does not exclude the case of dividing by zero.
expneg2 expineg2
expn1 expn1ap0
expcl2lem expcl2lemap
reexpclz reexpclzap
expclzlem expclzaplem
expclz expclzap
expne0 expap0
expne0i expap0i
expnegz expnegzap
mulexpz mulexpzap
exprec exprecap
expmulz expmulzap
expsub expsubap
expp1z expp1zap
expm1 expm1ap
expdiv expdivap
ltexp2 , leexp2 , leexp2 , ltexp2d , leexp2d none Presumably provable, but the set.mm proof uses ltord1
ltexp2r , ltexp2rd none Presumably provable, but the set.mm proof uses ltexp2
sqdiv sqdivap
sqgt0 sqgt0ap
sqrecii , sqrecd exprecap
sqdivi sqdivapi
sqgt0i sqgt0api
sqlecan lemul1 Unused in set.mm
sqeqori none The reverse direction is oveq1 together with sqneg. The forward direction is presumably not provable, see mul0or for more discussion. For the nonnegative real case see sq11.
subsq0i , sqeqor none Variations of sqeqori .
sq01 none Lightly used in set.mm. Presumably not provable as stated, for reasons analogous to mul0or .
crreczi none Presumably could be proved if not-equal is changed to apart, but unused in set.mm.
expmulnbnd none Should be possible to prove this or something similar, but the set.mm proof relies on case elimination based on whether 0 ≤ 𝐴 or not.
digit2 , digit1 none Depends on modulus and floor, and unused in set.mm.
modexp none Depends on modulus. Presumably it, or something similar, can be made to work as it is mostly about integers rather than reals.
discr1 , discr none The set.mm proof uses real number trichotomy.
sqrecd sqrecapd
expclzd expclzapd
exp0d expap0d
expne0d expap0d
expnegd expnegapd
exprecd exprecapd
expp1zd expp1zapd
expm1d expm1apd
expsubd expsubapd
sqdivd sqdivapd
expdivd expdivapd
reexpclzd reexpclzapd
sqgt0d sqgt0apd
mulsubdivbinom2 none Presumably provable if not equal is changed to apart.
muldivbinom2 none Presumably provable if not equal is changed to apart.
nn0le2msqi nn0le2msqd Although nn0le2msqi could be proved, having a version in deduction form will be more useful.
nn0opthlem1 nn0opthlem1d Although nn0opthlem1 could be proved, having a version in deduction form will be more useful.
nn0opthlem2 nn0opthlem2d Although nn0opthlem2 could be proved, having a version in deduction form will be more useful.
nn0opthi nn0opthd Although nn0opthi could be proved, having a version in deduction form will be more useful.
nn0opth2i nn0opth2d Although nn0opth2i could be proved, having a version in deduction form will be more useful.
facmapnn faccl Presumably provable now that iset.mm has df-map. But faccl would be sufficient for the uses in set.mm.
faclbnd4 , faclbnd5 , and lemmas none Presumably provable; unused in set.mm.
df-hash df-ihash
hashkf , hashgval , hashginv none Due to the differences between df-hash in set.mm and df-ihash here, there's no particular need for these as stated
hashinf hashinfom The condition that 𝐴 is infinite is changed from ¬ 𝐴 ∈ Fin to ω ≼ 𝐴.
hashbnd none The set.mm proof is not intuitionistic.
hashfxnn0 , hashf , hashxnn0 , hashresfn , dmhashres , hashnn0pnf none Although df-ihash is defined for finite sets and infinite sets, it is not clear we would be able to show this definition (or another definition) is defined for all sets.
hashnnn0genn0 none Not yet known whether this is provable or whether it is the sort of reverse closure theorem that we (at least so far) have been unable to intuitionize.
hashnemnf none Presumably provable but the set.mm proof relies on hashnn0pnf
hashv01gt1 hashfiv01gt1 It is not clear there would be any way to combine the finite and infinite cases.
hasheni hashen, hashinfom It is not clear there would be any way to combine the finite and infinite cases.
hasheqf1oi fihasheqf1oi It is not clear there would be any way to combine the finite and infinite cases.
hashf1rn fihashf1rn It is not clear there would be any way to combine the finite and infinite cases.
hasheqf1od fihasheqf1od It is not clear there would be any way to combine the finite and infinite cases.
hashcard none Cardinality is not well developed in iset.mm
hashxrcl none It is not clear there would be any way to combine the finite and infinite cases.
hashclb none Not yet known whether this is provable or whether it is the sort of reverse closure theorem that we (at least so far) have been unable to intuitionize.
nfile filtinf It is not clear there would be any way to combine the case where 𝐴 is finite and the case where it is infinite.
hashvnfin none This is a form of reverse closure, presumably not provable.
hashnfinnn0 hashinfom
isfinite4 isfinite4im
hasheq0 fihasheq0 It is not clear there would be any way to combine the finite and infinite cases.
hashneq0 , hashgt0n0 fihashneq0
hashelne0d none would be easy to prove if 𝐴𝑉 is changed to 𝐴 ∈ Fin
hashen1 fihashen1
hash1elsn none would be easy to prove if 𝐴𝑉 is changed to 𝐴 ∈ Fin
hashrabrsn none Presumably would need conditions around the existence of 𝐴 and decidability of 𝜑 but unused in set.mm.
hashrabsn01 none Presumably would need conditions around the existence of 𝐴 and decidability of 𝜑 but unused in set.mm.
hashrabsn1 none The set.mm proof uses excluded middle and this theorem is unused in set.mm.
hashfn fihashfn There is an added condition that the domain be finite.
hashgval2 none Presumably provable, when restated as (♯ ↾ ω) = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0), but lightly used in set.mm.
hashdom fihashdom There is an added condition that 𝐵 is finite.
hashdomi fihashdom It is presumably not possible to extend fihashdom beyond the finite set case.
hashun2 none The set.mm proof relies on undif2 (we just have undif2ss) and diffi (we just have diffifi)
hashun3 none The set.mm proof relies on various theorems we do not have
hashunx none It is not clear there would be any way to combine the finite and infinite cases.
hashge0 hashcl It is not clear there would be any way to combine the finite and infinite cases.
hashgt0 , hashge1 hashnncl It is not clear there would be any way to combine the finite and infinite cases.
hashnn0n0nn hashnncl To the extent this is reverse closure, we probably can't prove it. For inhabited versus non-empty, see fin0
elprchashprn2 hashsng Given either 𝑁 ∈ V or ¬ 𝑁 ∈ V this could be proved (as (♯‘{𝑀, 𝑁}) reduces to hashsng or hash0 respectively), but is not clear we can combine the cases (even 1domsn may not be enough).
hashprb hashprg
hashprdifel none This would appear to be a form of reverse closure.
hashle00 fihasheq0
hashgt0elex , hashgt0elexb fihashneq0 See fin0 for inhabited versus non-empty. It isn't clear it would be possible to also include the infinite case as hashgt0elex does.
hashss fihashss
hashin none Presumably additional conditions would be needed (see infi entry).
hashssdif fihashssdif
hashdif none Modified versions presumably would be provable, but this is unused in set.mm.
hashsn01 none Presumably not provable
hashsnle1 none At first glance this would appear to be the same as 1domsn but to apply fihashdom would require that the singleton be finite, which might imply that we cannot improve on hashsng.
hashsnlei none Presumably not provable
hash1snb , euhash1 , hash1n0 , hashgt12el , hashgt12el2 none Conjectured to be provable in the finite set case
hashsslei fihashss Would be provable if we transfered 𝐵 ∈ Fin from the conclusion to the hypothesis but as written falls afoul of ssfiexmid.
hashmap none Probably provable but the set.mm proof relies on a number of theorems which we don't have.
hashpw none Unlike pw2en this is only for finite sets, so it presumably is provable. The set.mm proof may be usable.
hashfun none Presumably provable but the set.mm proof relies on excluded middle and undif.
hashres , hashreshashfun none Presumably would need to add 𝐵 ∈ Fin or similar conditions.
hashimarn , hashimarni none Presumably would need an additional condition such as 𝐹 ∈ Fin but unused in set.mm.
fnfz0hashnn0 none Presumably would need an additional condition such as 𝑁 ∈ ℕ0 but unused in set.mm.
fnfzo0hashnn0 none Presumably would need an additional condition such as 𝑁 ∈ ℕ0 but unused in set.mm.
hashbc none The set.mm proof uses pwfi and ssfi .
hashf1 none The set.mm proof uses ssfi , excluded middle, abn0 , diffi and perhaps other theorems we don't have.
hashfac none The set.mm proof uses hashf1 .
leiso none The set.mm proof uses isocnv3 .
fz1iso zfz1iso The set.mm proof of fz1iso depends on OrdIso. Furthermore, trichotomy rather than weak linearity would seem to be needed.
ishashinf none May be possible with the antecedent changed from ¬ 𝐴 ∈ Fin to ω ≼ 𝐴 but the set.mm proof does not work as is.
phphashd , phphashrd none would appear to need hashclb or some other way of showing that the subset is finite
seqcoll seq3coll The functions 𝐹 and 𝐻 need to be defined on 𝑀 not just a subset thereof.
seqcoll2 none Presumably can be done with modifications similar to seq3coll.
df-word and all theorems relating to words over a set none presumably mostly provable
seqshft seq3shft
df-sgn and theorems related to the sgn function none To choose a value near zero requires knowing the argument with unlimited precision. It would be possible to define for rational numbers, or real numbers apart from zero.
mulre mulreap
rediv redivap
imdiv imdivap
cjdiv cjdivap
sqeqd none The set.mm proof is not intuitionistic, and this theorem is unused in set.mm.
cjdivi cjdivapi
cjdivd cjdivapd
redivd redivapd
imdivd imdivapd
df-sqrt df-rsqrt See discussion of complex square roots in the comment of df-rsqrt. Here's one possibility if we do want to define square roots on (some) complex numbers: It should be possible to define the complex square root function on all complex numbers satisfying (ℑ‘𝑥) # 0 ∨ 0 ≤ (ℜ‘𝑥), using a similar construction to the one used in set.mm. You need the real square root as a basis for the construction, but then there is a trick using the complex number x + |x| (see sqreu) that yields the complex square root whenever it is apart from zero (you need to divide by it at one point IIRC), which is exactly on the negative real line. You can either live with this constraint, which gives you the complex square root except on the negative real line (which puts a hole at zero), or you can extend it by continuity to zero as well by joining it with the real square root. The disjunctive domain of the resulting function might not be so useful though.
sqrtval sqrtrval See discussion of complex square roots in the comment of df-rsqrt
01sqrex and its lemmas resqrex Both set.mm and iset.mm prove resqrex although via different mechanisms so there is no need for 01sqrex.
cnpart none See discussion of complex square roots in the comment of df-rsqrt
sqrmo rsqrmo See discussion of complex square roots in the comment of df-rsqrt
resqreu rersqreu Although the set.mm theorem is primarily about real square roots, the iset.mm equivalent removes some complex number related parts.
sqrtneg , sqrtnegd none Although it may be possible to extend the domain of square root somewhat beyond nonnegative reals without excluded middle, in general complex square roots are difficult, as discussed in the comment of df-rsqrt
sqrtm1 none Although it may be possible to extend the domain of square root somewhat beyond nonnegative reals without excluded middle, in general complex square roots are difficult, as discussed in the comment of df-rsqrt
absrpcl , absrpcld absrpclap, absrpclapd
absdiv , absdivzi , absdivd absdivap, absdivapzi, absdivapd
absor , absori , absord qabsor It also would be possible to prove this for real numbers apart from zero, if we wanted
absmod0 none See df-mod ; we may want to supply this for rationals or integers
absexpz absexpzap
absz none Although this is presumably provable, the set.mm proof is not intuitionistic and it is lightly used in set.mm
recval recvalap
absgt0 , absgt0i absgt0ap, absgt0api
absmax maxabs
abs1m none Because this theorem provides (∗‘𝐴) / (abs‘𝐴) as the answer if 𝐴 ≠ 0 and 𝑖 as the answer if 𝐴 = 0, and uses excluded middle to combine those cases, it is presumably not provable as stated. We could prove the theorem with the additional condition that 𝐴 # 0, but it is unused in set.mm.
abslem2 none Although this could presumably be proved if not equal were changed to apart, it is lightly used in set.mm.
rddif , absrdbnd none If there is a need, we could prove these for rationals or real numbers apart from any rational. Alternately, we could prove a result with a slightly larger bound for any real number.
rexuzre none Unless the real number 𝑗 is known to be apart from an integer, it isn't clear there would be any way to prove this (see the steps in the set.mm proof which rely on the floor of a real number). It is unused in set.mm for whatever that is worth.
caubnd none If we can prove fimaxre3 it would appear that the set.mm proof would work with small changes (in the case of the maximum of two real numbers, using maxle1, maxle2, and maxcl).
sqreulem , sqreu , sqrtcl , sqrtcld , sqrtthlem , sqrtf , sqrtth , sqsqrtd , msqsqrtd , sqr00d , sqrtrege0 , sqrtrege0d , eqsqrtor , eqsqrtd , eqsqrt2d rersqreu, resqrtcl, resqrtcld, resqrtth As described at df-rsqrt, square roots of complex numbers are in set.mm defined with the help of excluded middle.
df-limsup and all superior limit theorems none This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to sequence convergence, supremums, etc.
df-rlim and theorems related to limits of partial functions on the reals none This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to convergence.
df-o1 and theorems related to eventually bounded functions none This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to sequence convergence, supremums, etc.
df-lo1 and theorems related to eventually upper bounded functions none This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to sequence convergence, supremums, etc.
reccn2 reccn2ap
isershft iser3shft
isercoll and its lemmas, isercoll2 none yet The set.mm proof would need modification
climsup none To show convergence would presumably require a hypothesis related to the rate of convergence.
climbdd none Presumably could be proved but the current proof of caubnd would need at least some minor adjustments.
caurcvg2 climrecvg1n
caucvg climcvg1n
caucvgb climcaucn, climcvg1n Without excluded middle, there are additional complications related to the rate of convergence.
iseralt none The set.mm proof relies on caurcvg2 which does not specify a rate of convergence.
df-sum df-sumdc The iset.mm definition/theorem adds a decidability condition and an if expression (which is to deal with differences in using seq for finite sums). It does function similarity to the set.mm definition of Σ.
sumex fsumcl, isumcl
sumeq2w sumeq2 Presumably could be proved, and perhaps also would rely only on extensionality (and logical axioms). But unused in set.mm.
sumeq2ii sumeq2
sum2id none The set.mm proof does not work as-is. Lightly used in set.mm.
sumfc sumfct
fsumcvg fsum3cvg
sumrb sumrbdc
summo summodc
zsum zsumdc
fsum fsum3
sumz isumz
sumss isumss
sumss2 isumss2
fsumcvg2 fsum3cvg2
fsumsers fsumsersdc
fsumcvg3 fsum3cvg3
fsumser fsum3ser
fsummsnunz none Could be proved if we added a 𝑍 ∈ V condition, but unused in set.mm.
isumdivc isumdivapc Changes not equal to apart
fsumcom2 fisumcom2 Although it is possible that (𝜑𝑘𝐶) → 𝐷 ∈ Fin can be proved from the other hypotheses, the set.mm proof of that uses ssfi .
fsum0diag fisum0diag Adds a 𝑁 ∈ ℤ hypothesis
fsumrev2 fisumrev2 Adds 𝑀 ∈ ℤ and 𝑁 ∈ ℤ hypotheses
fsum0diag2 fisum0diag2 Adds a 𝑁 ∈ ℤ hypothesis
fsumdivc fsumdivapc Changes not equal to apart
fsumless fsumlessfi Whether this can be proved without the 𝐶 ∈ Fin condition is unknown but such a proof would be fairly different from the set.mm proof.
seqabs fsumabs Finite sums are more naturally expressed with Σ rather than seq especially in iset.mm. Use fsum3ser as needed.
cvgcmp cvgcmp2n The set.mm proof of cvgcmp relies on caurcvg2 which does not specify a rate of convergence.
cvgcmpce none The proof, and perhaps the statement of the theorem, would need some changes related to the rate of convergence.
abscvgcvg none The set.mm proof relies on cvgcmpce
climfsum none Likely provable, but lightly used in set.mm.
qshash none The set.mm proof will not work as-is.
ackbijnn none iset.mm does not have ackbij1
incexclem , incexc , incexc2 none A metamath 100 theorem but otherwise unused in set.mm.
isumless isumlessdc Adds a decidability condition on the index set for the sum
isumsup2 , isumsup none Having an upper bound on the partial sums would not suffice; a stronger convergence condition would be needed.
isumltss none Should be provable with the addition of a decidability condition such as the one found in isumss2 and fsum3cvg3.
climcnds none The set.mm proof will not work without modifications.
divcnvshft none The set.mm proof uses ovex to show that 𝐴 / (𝑘 + 𝐵) is a set, even when 𝑘 + 𝐵 might be zero. This could be solved by adding another usage of df-div or proving (1 / 0) = ∅ but relying on the value of dividing by zero is not something we usually let ourselves do. Another solution would be to add a 0 < (𝑀 + 𝐵) hypothesis.
supcvg none The set.mm proof uses countable choice and also various supremum theorems proved via excluded middle.
infcvgaux1i , infcvgaux2i none See supcvg entry
harmonic none Should be feasible once we get isumless and climcnds (or similar theorems). A Metamath 100 theorem but otherwise unused in set.mm.
geoserg geosergap Not equal is changed to apart
geoser geoserap Not equal is changed to apart
pwm1geoser pwm1geoserap1 Adds a condition that the base is apart from one. The set.mm proof relies on case elimination on whether the base is one or not equal to one.
geomulcvg none The set.mm proof relies on cvgcmpce and expmulnbnd and would appear to also require 𝐴 to be apart from zero.
cvgrat cvgratgt0 Adds a 0 < 𝐴 condition which presumably is omitted from the set.mm theorem only for convenience (the theorem isn't interesting unless it holds).
mertens mertensabs Because we don't (yet at least) have abscvgcvg or anything else relating the convergence of a sequence's absolute values to the convergence of the sequence itself, we add the condition that both the sequence 𝐹 and the sequence of its absolute values converge (that is, seq0( + , 𝐹) ∈ dom ⇝ is an additional hypothesis beyond what set.mm has).
clim2div clim2divap
prodfmul prod3fmul The functions 𝐹, 𝐺, and 𝐻 need to be defined on (ℤ𝑀) not merely (𝑀...𝑁).
prodfn0 prodfap0
prodfrec prodfrecap
prodfdiv prodfdivap in addition to changing not equal to apart, various hypotheses need to hold for (ℤ𝑀) not merely (𝑀...𝑁)
ntrivcvg ntrivcvgap
ntrivcvgn0 ntrivcvgap0
ntrivcvgfvn0 none To be useful, presumably not equal needs to be changed to apart. This means the set.mm proof does not work and we need another approach (for example, something a bit like seq3p1 or seq3split but for convergence expressed with rather than for a finite subsequence).
ntrivcvgtail none the set.mm proof depends on ntrivcvgfvn0 . Would need not equal changed to apart.
ntrivcvgmullem , ntrivcvgmul none the set.mm proof depends on ntrivcvgtail . Would need not equal changed to apart.
df-prod df-proddc Changes not equal to apart, adds a decidability condition for indexing by a set of integers, and passes a more fully defined sequence to seq in the finite case. Should function similarly to the set.mm definition of .
prodex none presumably will need to be replaced by fprodcl and iprodcl analogous to sumex
prodeq2ii prodeq2 the set.mm proof uses excluded middle
prod2id none the set.mm proof uses prodeq2ii
prodrblem prodrbdclem
fprodcvg fproddccvg
prodrblem2 prodrbdclem2
prodrb prodrbdc
prodmolem3 prodmodclem3
prodmolem2a prodmodclem2a
prodmolem2 prodmodclem2
prodmo prodmodc
zprod zproddc
iprod iprodap
zprodn0 zprodap0
iprodn0 iprodap0
fprod fprodseq
eftval eftvalcn Adds an easily satisfied condition.
fprodefsum none Presumably feasible once finite products are better developed.
tanval tanvalap
tancl , tancld tanclap, tanclapd
tanval2 tanval2ap
tanval3 tanval3ap
retancl , retancld retanclap, retanclapd
tanneg tannegap
sinhval , coshval , resinhcl , rpcoshcl , recoshcl , retanhcl , tanhlt1 , tanhbnd none yet should be provable
sinltx sin01bnd, sinbnd Although we can prove the 𝐴 ≤ 1 case (see sin01bnd) or the 1 < 𝐴 case (from sinbnd), set.mm uses real number trichotomy to combine those cases.
ruc none A proof would need either excluded middle or countable choice, per [BauerHanson]
dvdsaddre2b none Something along these lines (perhaps with real changed to rational) may be possible
fsumdvds , 3dvds none May be possible when summation is well enough developed
sumeven , sumodd , evensumodd , oddsumodd , pwp1fsum , oddpwp1fsum none Presumably possible when summation is well enough developed
divalglem0 and other divalg lemmas divalglemnn and other lemmas Since the end result divalg is the same, we don't list all the differences in lemmas here.
gcdcllem1 , gcdcllem2 , gcdcllem3 gcdn0cl, gcddvds, dvdslegcd These are lemmas which are part of the proof of theorems that iset.mm proves a somewhat different way
seq1st none The sequence passed to seq, at least as handled in theorems such as seqf, must be defined on all integers greater than or equal to 𝑀, not just at 𝑀 itself. It may be possible to patch this up, but seq1st is unused in set.mm.
algr0 ialgr0 The 𝐹:𝑆𝑆 hypothesis is added (related theorems already have that hypothesis).
df-lcmf and theorems using it none Although this could be defined, most of the theorems would need decidability conditions analogous to zsupcl
absproddvds , absprodnn none Needs product to be developed, but once that is done seems like it might be possible.
fissn0dvds , fissn0dvdsn0 none Possibly could be proved using findcard2 or the like.
coprmprod , coprmproddvds none Can investigate once product is better developed.
isprm5 none Presumably provable, but the set.mm proof relies on excluded middle in multiple places.
isprm7 none The set.mm proof relies on isprm5
maxprmfct none Presumably provable with suitable adjustments to the condition for the existence of the supremum
ncoprmlnprm none Presumably provable but the set.mm proof uses excluded middle
zsqrtelqelz nn0sqrtelqelz We don't yet have much on the square root of a negative number
df-odz and all theorems concerning the order function on the class of integers mod N none Presumably could be defined, but would require changes to how we show the infimum exists. Lightly used in set.mm.
eulerth none The lemma eulerthlem2 relies on seqf1o
fermltl none Relies on eulerth
prmdiv none Relies on eulerth
prmdiveq none Relies on prmdiv
prmdivdiv none Relies on prmdiveq
phisum none May be provable once summation is better developed
unben none not possible as stated, as shown by exmidunben
isstruct2 isstruct2im, isstruct2r The difference is the addition of the 𝐹𝑉 condition for the reverse direction.
isstruct isstructim, isstructr The difference is the addition of the 𝐹𝑉 condition for the reverse direction.
slotfn slotslfn
strfvnd strnfvnd
strfvn strnfvn
strfvss strfvssn
setsval setsvala
setsidvald strsetsid
fsets none Apparently would need decidable equality on 𝐴 or some other condition.
setsdm none The set.mm proof relies on undif1
setsstruct2 none The set.mm proof relies on setsdm
setsexstruct2 , setsstruct none The set.mm proofs rely on setsstruct2
setsres setsresg
setsabs setsabsd
strfvd strslfvd
strfv2d strslfv2d
strfv2 strslfv2
strfv strslfv
strfv3 strslfv3
str0 strsl0
strfvi none The set.mm proof uses excluded middle to combine the proper class and set cases.
setsid setsslid
setsnid setsslnid
sbcie2s none Apparently would require conditions that 𝐴 and 𝐵 are sets.
sbcie3s none Apparently would require conditions that 𝐴, 𝐵, and 𝐶 are sets.
elbasfv , elbasov , strov2rcl none The set.mm proofs rely on excluded middle.
basprssdmsets none The set.mm proof relies on setsdm
ressval ressid2, ressval2 For the 𝐵𝐴 and ¬ 𝐵𝐴 cases, respectively.
ressbas none Apparently needs to have conditions added, for example that 𝑊 is a set plus one of 𝐵𝐴 or ¬ 𝐵𝐴.
ressbas2 , ressbasss none The set.mm proof relies on ressbas .
resslem , ress0 none The set.mm proof relies on excluded middle.
ressinbas none Apparently needs to have conditions added, for example that 𝑊 is a set plus one of 𝐵𝐴 or ¬ 𝐵𝐴.
ressval3d none The set.mm proof relies on setsidvald and sspss .
ressress none The set.mm proof relies on ressinbas and excluded middle.
ressabs none The set.mm proof relies on ressress but at first glance this would appear to be feasible given the 𝐵𝐴 condition.
strle1 strle1g
strle2 strle2g
strle3 strle3g
1strstr 1strstrg
2strstr 2strstrg
2strbas 2strbasg
2strop 2stropg
2strstr1 2strstr1g
2strbas1 2strbas1g
2strop1 2strop1g
grpstr grpstrg
grpbase grpbaseg
grpplusg grpplusgg
ressplusg none The set.mm proof relies on resslem
grpbasex , grpplusgx grpbaseg, grpplusgg Marked as discouraged even in set.mm.
rngstr rngstrg
rngbase rngbaseg
rngplusg rngplusgg
rngmulr rngmulrg
ressmulr , ressstarv none The set.mm proof relies on resslem
srngstr srngstrd
srngbase srngbased
srngplusg srngplusgd
srngmulr srngmulrd
srnginvl srnginvld
lmodstr lmodstrd
lmodbase lmodbased
lmodplusg lmodplusgd
ipsstr ipsstrd
ipsbase ipsbased
ipsmulr ipsmulrd
ipsip ipsipd
resssca , ressvsca , ressip none The set.mm proof relies on resslem
phlstr , phlbase , phlplusg , phlsca , phlvsca , phlip none Intuitionizing these will be straightforward once we get around to it, in a manner similar to lmodstrd. The proofs will use theorems like strle1g, strleund, and opelstrsl.
topgrpstr topgrpstrd
topgrpbas topgrpbasd
topgrpplusg topgrpplusgd
topgrptset topgrptsetd
resstset none The set.mm proof relies on resslem
otpsstr , otpsbas , otpstset , otpsle none Unused in set.mm. If we want to develop this more we may need to figure out whether to define order in terms of < or as the relationship between those may be different without excluded middle.
0rest none Might need a 𝐴 ∈ V condition added, and this theorem seems to be mostly be used in conjunction with excluded middle.
topnval topnvalg
topnid topnidg
topnpropd topnpropgd
prdsbasex none Would need some conditions on whether 𝑅 is a function, on set existence, or the like. However, it is unused in set.mm.
imasvalstr , prdsvalstr , prdsvallem , prdsval , prdssca , prdsbas , prdsplusg , prdsmulr , prdsvsca , prdsip , prdsle , prdsless , prdsds , prdsdsfn , prdstset , prdshom , prdsco , prdsbas2 , prdsbasmpt , prdsbasfn , prdsbasprj , prdsplusgval , prdsplusgfval , prdsmulrval , prdsmulrfval , prdsleval , prdsdsval , prdsvscaval , prdsvscafval , prdsbas3 , prdsbasmpt2 , prdsbascl , prdsdsval2 , prdsdsval3 , pwsval , pwsbas , pwselbasb , pwselbas , pwsplusgval , pwsmulrval , pwsle , pwsleval , pwsvscafval , pwsvscaval , pwssca , pwsdiagel , pwssnf1o none At a minimum, these theorems would need new set existence conditions and other routine intuitionizing. At worst, they would need a bigger revamp for things like how order works.
df-ordt , df-xrs , and all theorems involving ordTop or RR*s none The set.mm definitions would not seem to fit with constructive definitions of these concepts (for example, the if in df-xrs would apparently needed to be expressed in terms of a suitable maximum and perhaps other changes are needed too)
df-qtop , df-imas , df-qus and all theorems defined in terms of quotient topology, image structure, and quotient ring. none presumably could be added in some form
df-xps and all theorems mentioning the binary product on a structure (syntax Xs.) none The set.mm definition depends on df-imas ( "s ) and df-prds ( Xs_ ). By its nature the definition of the binary product considers every structure slot (including most notably order which presumably will need to be handled differently in iset.mm).
df-mre , df-mrc and all theorems using the Moore or mrCls syntax none The closest we have currently is df-cls but even that doesn't function as it does in set.mm (because complements are different without excluded middle).
df-cnfld and all theorems using CCfld none Could presumably be defined in some form, but we'd have to look at the literature definitions of a constructive field and see how we'd need to define this. Plus questions about whether to define order in terms of < or .
istop2g istopfin
iinopn none The set.mm proof relies on abrexfi
riinopn none The set.mm proof relies on iinopn (the other issues in the set.mm proof could apparently be handled by fin0or).
rintopn none The set.mm proof relies on riinopn
toponsspwpw toponsspwpwg
toprntopon none Presumably could be proved but the set.mm proof does not work as it is.
topsn none The set.mm proof relies on pwsn
tpsprop2d none The set.mm proof does not work as-is. Unused in set.mm
basdif0 none The set.mm proof relies on undif1
0top none The set.mm proof relies on sssn
en2top none The set.mm proof relies on strict dominance
2basgen 2basgeng
tgdif0 none The set.mm proof relies on excluded middle
indistopon none The set.mm proof relies on sspr
indistop , indisuni none The set.mm proof relies on indistopon
fctop none The set.mm proof relies on theorems we don't have including con1d , ssfi , unfi , rexnal , and difindi .
fctop2 none The set.mm proof relies on fctop
cctop none The set.mm proof relies on theorems we don't have including con1d , rexnal , and difindi .
ppttop , pptbas none The set.mm proof relies on theorems we don't have including orrd and ianor
indistpsx , indistps , indistps2 , indistpsALT , indistps2ALT none The set.mm proofs rely on indiscrete topology theorems we don't have
isopn2 none The set.mm proof relies on dfss4
opncld none The set.mm proof relies on isopn2
iincld none The set.mm proof relies on opncld
intcld none The set.mm proof relies on iincld
incld none The set.mm proof relies on intcld
riincld none The set.mm proof relies on iincld , incld , and case elimination
clscld none The set.mm proof relies on intcld
clsf none The set.mm proof relies on clscld
clsval2 none The set.mm proof relies on dfss4 and opncld
ntrval2 none The set.mm proof relies on dfss4 and clsval2
ntrdif , clsdif none
cmclsopn none The set.mm proof relies on dfss4 and clsval2
cmntrcld none The set.mm proof relies on opncld
iscld3 , iscld4 none
clsidm none The set.mm proof relies on clscld
0ntr none The set.mm proof relies on ssdif0
elcls , elcls2 none
clsndisj none The set.mm proof relies on elcls
elcls3 none The set.mm proof relies on elcls
opncldf1 , opncldf2 , opncldf3 none The set.mm proofs rely on dfss4 and opncld
isclo none One direction of the biconditional may be provable by taking the set.mm proof and replacing undif with undifss
isclo2 none The set.mm proof relies on isclo
indiscld none Something along these lines may be possible once we define the indiscrete topology
neips neipsm
neindisj none The set.mm proof relies on clsndisj
opnnei none Apparently the set.mm proof could easily be adapted for the case in which 𝑆 is inhabited
neindisj2 none The set.mm proof relies on elcls
neipeltop , neiptopuni , neiptoptop , neiptopnei , neiptopreu none The set.mm proofs rely on several theorems we do not have.
df-lp and all theorems using the limPt syntax none
df-perf and all theorems using the Perf syntax none
restbas restbasg
restsn2 none The set.mm proof relies on topsn
restcld none The set.mm proof relies on opncld
restcldi none The set.mm proof relies on restcld
restcldr none The set.mm proof relies on restcld
restfpw none The set.mm proof relies on ssfi
neitr none The set.mm proof relies on inundif
restcls none The set.mm proof relies on clscld
restntr none The set.mm proof relies on excluded middle
resstopn none The set.mm proof relies on resstset
resstps none The set.mm proof relies on resstopn
lmrel lmreltop
iscnp2 iscnp
cnptop1 , cnptop2 none
cnprcl cnprcl2k
cnpf , cnpcl cnpf2
cnprcl2 cnprcl2k
cnpimaex icnpimaex
cnpco cnptopco
iscncl none The set.mm proof relies on opncld
cncls2i none The set.mm proof relies on clscld
cnclsi none The set.mm proof relies on cncls2i
cncls2 none The set.mm proof relies on cncls2i and iscncl
cncls none The set.mm proof relies on cncls2 and cnclsi
cncnp2 cncnp2m
cnpresti cnptopresti
cnprest cnptoprest
cnprest2 cnptoprest2
cnindis none The set.mm proof relies on indiscrete topology theorems that we don't have.
paste none The set.mm proof relies on restcldr
lmcls , lmcld none The set.mm proof relies on elcls
lmcnp lmtopcnp
df-cmp and all compactness (syntax Comp) theorems none How compactness fares without excluded middle is a complicated topic. See for example [HoTT], section 11.5.
df-haus and all Hausdorff (syntax Haus) theorems none Perhaps there would need to be an apartness relation to replace the use of negated equality.
df-conn and all connected toplogy (syntax Conn) theorems none would apparently need a lot of changes to work
df-1stc and all first-countable theorems none Worth considering the definition of countable seen in theorems such as ctm and finct, as well as whatever else might come up.
df-2ndc and all second-countable theorems none Worth considering the definition of countable seen in theorems such as ctm and finct, as well as whatever else might come up.
df-lly and all theorems using the Locally syntax none
df-xko and all theorems using the compact-open topology syntax (^ko) none not clear what is possible here
ptval none This would need more extensive development of theorems related to the Xt_ syntax (not just df-pt itself).
ptpjpre1 none Perhaps would be provable in the case where 𝐴 has decidable equality.
elpt , elptr , elptr2 , ptbasid , ptuni2 , ptbasin , ptbasin2 , ptbas , ptpjpre2 , ptbasfi , pttop , ptopn , ptopn2 none Although some parts of these product topology theorems may be intuitionizable, it isn't clear doing so would produce a set of theorems which function as desired.
txcld none The set.mm proof relies on difxp
txcls none The set.mm proof relies on txcld
txcnpi none Should be provable (via icnpimaex and cnpf2) but may need an additional 𝐿 ∈ Top hypothesis.
ptcld none The set.mm proof depends on pttop , boxcutc , ptopn2 , and riincld
dfac14 none The left hand side implies excluded middle by acexmid; we could see whether the proof that the right hand side implies choice is also valid without excluded middle.
txindis none The set.mm proof relies on excluded middle, indistop , and indisuni .
df-hmph and all theorems using that syntax none presumably would need some modifications to the parts which use set difference
hmeofval hmeofvalg
hmeocls none the set.mm proof uses cncls2i
hmeoqtop none relies on qTop syntax
pt1hmeo none the set.mm proof uses ptcn
ptuncnv , ptunhmeo none the set.mm proofs use ptuni
xmetrtri2 none Presumably this or something similar could be defined once we define RR*s ( df-xrs ) or something along those lines.
xmetgt0 , metgt0 xmeteq0, meteq0 Presumably would require defining apartness on 𝑋 or something along those lines.
xbln0 xblm
blin blininf
setsmsbas setsmsbasg
setsmsds setsmsdsg
setsmstset setsmstsetg
setsmstopn , setsxms , setsms , tmsval , tmslem , tmsbas , tmsds , tmstopn , tmsxms , tmsms none Presumably could prove these or similar theorems, analogous to setsmsbasg
blcld none The set.mm proof relies on xrltnle
blcls none The set.mm proof relies on blcld
blsscls none The set.mm proof relies on blcls
stdbdmetval bdmetval
stdbdxmet bdxmet
stdbdmet bdmet
stdbdbl bdbl
stdbdmopn bdmopn
ressxms , ressms none Awaits revision of df-ress as described at Clean up multifunction restriction operator for extensible structures in iset.mm
prdsxms , prdsms none This would need more extensive development of theorems related to the Xs_ syntax (not just df-prds itself).
pwsxms , pwsms none This would need more extensive development of theorems related to the ^s syntax (not just df-pws itself).
tmsxps xmetxp tmsxps relies on structure products and related theorems
tmsxpsmopn xmettx tmsxpsmopn relies on structure products and related theorems
dscmet , dscopn none The set.mm definition for the discrete metric, and the proofs, would seem to rely on equality being decidable.
qtopbaslem qtopbasss
iooretop iooretopg
icccld , icopnfcld , iocmnfcld , qdensere none depend on various closed set theorems
cnfldtopn none if we use (MetOpen‘(abs ∘ − )) as our notation for the topology of the complex numbers, this theorem is not needed
cnfldtopon cntoptopon
cnfldtop cntoptop
unicntop unicntopcntop
cnopn cnopncntop
qdensere2 none the set.mm proof depends on qdensere
blcvx none presumably provable for either the 𝑇 = 0 case or the 𝑇 ∈ (0(,]1) case; set.mm uses excluded middle to combine those two cases
tgioo2 tgioo2cntop
rerest rerestcntop
tgioo3 none Until we have defined RRfld (presumably closely related to the issues described at the df-cnfld entry here), we can use (topGen‘ran (,)) as a notation for the topology of the real numbers (as seen at tgioo2cntop).
zcld none the set.mm proof uses the floor function in ways that we are unlikely to be able to intuitionize
iccntr none the set.mm proof uses real number trichotomy in many steps
opnreen none the set.mm proof is not usable as-is but it would be interesting to see if some portions can be adapted
rectbntr0 none the set.mm proof relies on various theorems we do not have
xmetdcn2 none the set.mm theorem is defined in terms of the RR*s syntax
xmetdcn none the set.mm theorem is defined in terms of the ordTop syntax and uses xmetdcn2 in the proof
metdcn2 none the set.mm proof relies on xmetdcn
metdcn none the set.mm proof relies on metdcn2
msdcn none the set.mm proof relies on metdcn2
cnmpt1ds none the set.mm proof relies on msdcn
cnmpt2ds none the set.mm proof relies on msdcn
abscn abscn2, abscncf presumably provable (expressing the topology of the complex numbers as (MetOpen‘(abs ∘ − )) if iset.mm doesn't have CCfld yet).
metdsval none the set.mm proof uses infex
metdsf none the set.mm proof uses infxrcl
addcnlem addcncntoplem expresses the topology of the complex numbers as (MetOpen‘(abs ∘ − ))
addcn addcncntop expresses the topology of the complex numbers as (MetOpen‘(abs ∘ − ))
subcn subcncntop expresses the topology of the complex numbers as (MetOpen‘(abs ∘ − ))
mulcn mulcncntop expresses the topology of the complex numbers as (MetOpen‘(abs ∘ − ))
divcn divcnap
fsumcn fsumcncntop
fsum2cn none the set.mm proof uses fsumcn
expcn none the set.mm proof uses mulcn
divccn none Not equal would need to be changed to apart. Also, the set.mm proof uses mulcn
sqcn none the set.mm proof uses expcn
divccncf divccncfap
cncfcn cncfcncntop
cncfcn1 cncfcn1cntop
cncfmpt2f cncfmpt2fcntop
cncfmpt2ss none would apparently only need a change to the notation for the topology on the complex numbers
cdivcncf cdivcncfap
cnmpopc none this kind of piecewise definition would apparently rely on real number trichotomy or something similar
cnrehmeo cnrehmeocntop
ivth ivthinc
ivth2 ivthdec
ivthle2 ivthdec the set.mm proof uses leloe
df-limc df-limced df-limced is adapted from ellimc3 in set.mm
df-dv df-dvap The definition is adjusted for the notation of the topology on the complex numbers, and for apartness versus negated equality.
df-dvn and all theorems mentioning iterated derivative (Dn) none should be easily intuitionizable
df-cpn and all theorems mentioning -times continuously differentiable functions (C^n) none should be easily intuitionizable
reldv reldvg
limcvallem , limcfval , ellimc none rely on decidability of real number equality
ellimc3 ellimc3apf, ellimc3ap
limcdif limcdifap changes not equal to apart and adds 𝐴 ⊆ ℂ hypothesis
ellimc2 none Presumably provable with not equal changed to apart. If iset.mm doesn't have CCfld yet, use (MetOpen‘(abs ∘ − )) for the topology of the complex numbers (see cnfldtopn in set.mm).
limcmo limcimo
limcmpt limcmpted
limcmpt2 none
limcres none Although this would appear to be provable, it might benefit from some additional theorems which help us manipulate t and metric spaces. If iset.mm doesn't have CCfld yet, use (MetOpen‘(abs ∘ − )) for the topology of the complex numbers (see cnfldtopn in set.mm). Proof sketch: because interiors are open, we can form a ball around 𝐵 which is contained in ((int‘𝐽)‘(𝐶 ∪ {𝐵})) which gives us the delta we need to apply ellimc3ap for (𝐹 lim 𝐵) (given that we can apply ellimc3ap for ((𝐹𝐶) lim 𝐵) when working within 𝐶).
cnplimc cnplimccntop
limccnp limccnpcntop
limccnp2 limccnp2cntop
limcco limccoap limccoap is only usable in the case where 𝑥 # 𝑋 implies R(x) # 𝐶 so it is less general than limcco
limciun none The set.mm proof relies on fofi , rintopn and ellimc2 .
limcun none The set.mm proof relies on limciun .
dvlem dvlemap not equal is changed to apart
dvfval dvfvalap changes not equal to apart and changes the notation for the topology on the complex numbers
eldv eldvap changes not equal to apart and changes the notation for the topology on the complex numbers
dvbssntr dvbssntrcntop changes the notation for the topology on the complex numbers
dvbsss dvbsssg
dvfg dvfgg
dvf dvfpm
dvfcn dvfcnpm
dvres none The set.mm proof relies on restntr and limcres . It may be worth seeing if it is easier to prove the 𝑆 ∈ {ℝ, ℂ} case.
dvres2 none The set.mm proof relies on restntr
dvres3 none The set.mm proof relies on dvres2
dvres3a none The set.mm proof relies on dvres2
dvidlem dvidlemap
dvcnp none would appear to rely on decidable equality of real numbers
dvcnp2 dvcnp2cntop
dvaddbr dvaddxxbr dvaddbr allows 𝑋 and 𝑌 to be different and uses excluded middle when handling that possibility.
dvmulbr dvmulxxbr
dvmul dvmulxx
dvmulf dvimulf
dvcmul none unused in set.mm and the set.mm proof depends on dvres2
dvcmulf none the set.mm proof depends on dvres , dvres3 , and caofid2
dvcobr dvcoapbr dvcoapbr adds a 𝑢 # 𝐶 → (𝐺𝑢) # (𝐺𝐶) condition
dvco none the set.mm proof depends on dvcobr
dvcof none the set.mm proof depends on dvcobr and dvco
dvrec dvrecap
dvmptres3 none the set.mm proof depends on dvres3a
dvmptid dvmptidcn the version for real numbers would presumably be provable as well
dvmptc dvmptccn the version for real numbers would presumably be provable as well
dvmptcl dvmptclx adds a 𝑋𝑆 hypothesis
dvmptmul dvmptmulx adds a 𝑋𝑆 hypothesis
dvmptres2 none the set.mm proof uses dvres
dvmptres none the set.mm proof uses dvmptres2
dvmptcmul dvmptcmulcn dvmptcmulcn is the case where both 𝑋 and 𝑆 are itself rather than subsets. The set.mm proof of dvmptcmul uses dvmptres2 .
dvmptdivc none the set.mm proof uses dvmptcmul
dvmptneg dvmptnegcn dvmptnegcn is for rather than a subset thereof
dvmptsub dvmptsubcn dvmptsubcn is for rather than a subset thereof
dvmptcj dvmptcjx adds a 𝑋𝑆 hypothesis
dvmptre none Presumably needs a 𝑋 ⊆ ℝ condition added. Also, the set.mm proof relies on dvmptcmul (with 𝑆 set to ).
dvmptim none Presumably needs a 𝑋 ⊆ ℝ condition added. Also, the set.mm proof relies on dvmptcmul (with 𝑆 set to ).
dvmptntr none the set.mm proof uses dvres
dvmptco none the set.mm proof uses dvcof
dvrecg none the set.mm proof uses dvmptco
dvmptdiv none the set.mm proof uses dvrecg
dvmptfsum none the set.mm proof uses dvmptc and dvmptres
dvcnvlem , dvcnv none Not equal would need to change to apart and notation for the topology on the complex numbers would change. More seriously, the set.mm proof uses limcco . Changing to limccoap may be possible given a condition along the lines of 𝑥𝑋𝑦𝑋(𝑥 # 𝑦 ↔ (𝐹𝑥) # (𝐹𝑦).
dvexp3 none the set.mm proof uses dvmptres and dvmptco
dvle none the set.mm proof uses iccntr , dvmptntr , dvmptsub , and dvge0
dvcnvre none the set.mm proof depends on compactness ( icccmp and cncfcnvcn ), restntr , evthicc2 , dvres , iccntr , and dvne0
efcvx none the set.mm proof uses dvres , dvres3 , iccntr , and dvcvx
reefgim none
pilem2 pilem3 our proof of pilem3 is different enough from set.mm that it doesn't need to go via pilem2
tanabsge none the set.mm proof uses real number trichotomy
sinq12ge0 none the set.mm proof uses leloe
cosq14ge0 none the set.mm proof uses sinq12ge0
pige3ALT pige3 the set.mm proof uses various theorems we do not have
sineq0 none the set.mm proof uses various theorems not in iset.mm
coseq1 none the set.mm proof uses sineq0
efeq1 none the set.mm proof uses sineq0
cosne0 none the set.mm proof uses sineq0
cosord cosordlem, cos11 presumably provable but the set.mm proof uses real number trichotomy
sinord none the set.mm proof depends on cosord
recosf1o ioocosf1o the set.mm proof of recosf1o relies on ivthle2
resinf1o none the set.mm proof relies on recosf1o
tanord1 none the set.mm proof relies on ltord1
tanord none the set.mm proof relies on tanord1 , real number trichotomy, and ltord1
tanregt0 none the set.mm proof relies on cosne0 , rpcoshcl , rptanhcl , and tanhbnd
efgh none depends on SubGrp and CCfld
efif1o none depends on complex square root
efifo none depends on efif1o
eff1o , efabl , efsubm , circgrp , circsubm none
df-log df-relog
df-cxp df-rpcxp we only define the power function for a positive real base for reasons described in the df-relog comment
logrn , ellogrn , dflog2 none see df-relog comment for discussion of complex logarithms
relogrn none Should be provable given reeff1o but perhaps it is better to avoid the theorems involving ran log from set.mm because they would mean something different with df-relog than with the set.mm definition.
logrncn , eff1o2 , logf1o none see df-relog comment for discussion of complex logarithms
logrncl none see df-relog comment for discussion of complex logarithms
logcl relogcl
logimcl none see df-relog comment for discussion of complex logarithms
logcld relogcld
logimcld , logimclad , abslogimle , logrnaddcl none see df-relog comment for discussion of complex logarithms
eflog reeflog
logeq0im1 none see df-relog comment for discussion of complex logarithms
logccne0 logrpap0
logne0 logrpap0 would be provable but logrpap0 changes not equal to apart and will generally be more helpful)
logef relogef
logeftb relogeftb
logneg , logm1 , lognegb none See df-relog comment for discussion of complex logarithms (including logarithms on negative reals, since iset.mm only defines the logarithm on +).
explog reexplog
relog none see df-relog comment for discussion of complex logarithms
reloggim none may be possible if more group theorems and notation are available
logfac none Should be provable. Because of the way seqhomo is used in the set.mm proof, it may be necessary to special-case 𝑁 = 1 and start the main proof with two, or prove an analogue of seq3homo which allows using + on one side of the equation and on the other.
eflogeq none see df-relog comment for discussion of complex logarithms
logcj , efiarg , cosargd , cosarg0d , argregt0 , argrege0 , argimgt0 , argimlt0 , logimul , logneg2 , logmul2 , logdiv2 , abslogle , tanarg none see df-relog comment for discussion of complex logarithms
logdivlt none the set.mm proof uses real number trichotomy
logdivle none the set.mm proof uses logdivlt
dvrelog none the set.mm proof uses dvres3 and dvcnvre
relogcn none the set.mm proof uses dvrelog
ellogdm , logdmn0 , logdmnrp , logdmss , logcnlem2 , logcnlem3 , logcnlem4 , logcnlem5 , logcn , dvloglem , logdmopn , logf1o2 , dvlog , dvlog2lem , dvlog2 none see df-relog comment for discussion of complex logarithms
advlog none the set.mm proof uses dvmptid , dvmptres , dvrelog , dvmptc , and dvmptsub
advlogexp none the set.mm proof uses various derivative theorems not present in iset.mm
efopn none the set.mm proof makes use of the complex logarithm
logtayl , logtaylsum , logtayl2 none see df-relog comment for discussion of complex logarithms
logccv none the set.mm proof uses dvrelog , dvmptneg , dvmptres2 , dvcvx , and soisoi
cxpval , cxpef , cxpefd rpcxpef as described at df-rpcxp we only support positive real bases
0cxp , 0cxpd none as described at df-rpcxp we only support positive real bases
cxpexpz , cxpexp , cxpexpzd cxpexpnn, cxpexprp as described at df-rpcxp we only support positive real bases
cxp0 , cxp0d rpcxp0, rpcxp0d as described at df-rpcxp we only support positive real bases
cxp1 , cxp1d rpcxp1, rpcxp1d as described at df-rpcxp we only support positive real bases
cxpcl , cxpcld rpcncxpcl, rpcncxpcld as described at df-rpcxp we only support positive real bases
recxpcl , recxpcld rpcxpcl as described at df-rpcxp we only support positive real bases
cxpne0 , cxpne0d cxpap0 the base must be a positive real and the result changes not equal to apart
cxpeq0 none since we do not support a mantissa of zero, does not apply
cxpp1 , cxpp1d rpcxpp1
cxpneg , cxpnegd rpcxpneg
cxpsub , cxpsubd rpcxpsub
cxpge0 , cxpge0d rpcxpcl we only support positive real bases
mulcxp , mulcxpd rpmulcxp
divcxp , divcxpd rpdivcxp
cxpmul2 , cxpmul2d cxpmul plus cxpexprp
cxproot rpcxproot
cxpmul2z , cxpmul2zd cxpmul plus cxpexprp
abscxp2 abscxp
cxplea , cxplead none the set.mm proof uses leloe
cxple2 , cxple2d rpcxple2
cxplt2 , cxplt2d rpcxplt2
cxple2a , cxple2ad none the set.mm proof uses leloe
cxpsqrt rpcxpsqrt
cxpsqrtth rpcxpsqrtth
dvcxp1 none the set.mm proof uses dvrelog , dvmptco , and dvmptres3
dvcxp2 none the set.mm proof uses dvmptco
dvsqrt none the set.mm proof uses dvcxp1
dvcncxp1 none this is for complex bases, see dvcxp1 entry concerning a positive real base
dvcnsqrt none see df-rsqrt comment for discussion of complex square roots
cxpcn none as described at df-rpcxp we only support positive real bases
cxpcn2 none presumably provable; the obvious proof would be via relogcn
cxpcn3 none as described at df-rpcxp we only support positive real bases
resqrtcn none the set.mm proof is in terms of the complex exponential, but this theorem should be provable one way or another
sqrtcn none see df-rsqrt comment for discussion of complex square roots
cxpaddle none the set.mm proof relies on leloe and other theorems we do not have
abscxpbnd rpabscxpbnd
root1id , root1eq1 , root1cj , cxpeq none as described at df-rpcxp we only support positive real bases
loglesqrt none presumably provable, but the set.mm proof relies on dvmptc , dvmptres , dvrelog , dvmptco , dvle , and relogcn
logrec none see df-relog comment for discussion of complex logarithms
logbval rplogbval
logbcl rplogbcl
logbid1 rplogbid1
logb1 rplogb1
elogb rpelogb
logbchbase rplogbchbase
relogbcl rplogbcl
relogbreexp rplogbreexp
relogbzexp rplogbzexp
relogbmul rprelogbmul
relogbmulexp rprelogbmulexp
relogbdiv rprelogbdiv
relogbexp relogbexpap
relogbcxp rplogbcxp
cxplogb rpcxplogb
relogbcxpb relogbcxpbap
logbmpt , logbf , logbfval , relogbf , logblog none we could add a version of these if iset.mm gets the curry syntax
irrdiff apdiff

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