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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) → (𝜓 → 𝜑)) (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.

Axioms of intuitionistic propositional calculus

Axiom Simp	ax-1	⊢ (φ → (ψ → φ))
Axiom Frege	ax-2	⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)))
Rule of Modus Ponens	ax-mp	|- ph   &   |- ph -> ps   ⇒   |- ps
Left 'and' elimination	ax-ia1	|- ( ( ph /\ ps ) -> ph )
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, φ ∨ ψ is not equivalent to ¬ φ → ψ, nor is φ → ψ equivalent to ¬ φ ∨ ψ, nor is φ ∧ ψ 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.

Axioms of intuitionistic predicate logic

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 e. 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.

Intuitionistic Zermelo-Fraenkel Set Theory (IZF)

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

Praeclarum theorema

Contraposition introduction

Double negation introduction

Triple negation

Definition of exclusive or

Negation and the false constant

Predicate calculus

Existential and universal quantifier swap

Existentially quantified implication

x = x

Definition of proper substitution

Double existential uniqueness

Set theory

Commutative law for union

A basic relationship between class and wff variables

Two ways of saying "is a set"

The ZF axiom of foundation implies excluded middle

Russell's paradox

Ordinal trichotomy implies excluded middle

Mathematical (finite) induction

Peano's postulates for arithmetic: 1 2 3 4 5

Two natural numbers are either equal or not equal (a special case of the law of the excluded middle which we can prove).

A natural number is either zero or a successor

The axiom of choice implies excluded middle

Burali-Forti paradox

Transfinite induction

Closure law for ordinal addition

Real and complex numbers

Archimedean property of real numbers

Properties of apartness: 1 2 3 4

The square root of 2 is irrational (a different statement than The square root of 2 is not rational)

Convergence of a sequence of complex numbers given a condition on the rate of convergence

Triangle inequality for absolute value

The maximum of two real numbers

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 e. _V and -. A e. _V you probably just care about the A e. _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 A. and then get E. 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 ph \/ -. ph 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. mo2 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, (/) e. A and A =/= (/) are equivalent (where A is an ordinal). In such theorems, (/) e. 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 A e. N. are added, or set.mm's ltbtwnnq versus iset.mm's ltbtwnnqq, in which E. x is changed to E. x e. Q.). If the result of applying a function is inhabited, then we know we applied it within its domain - that is relelfvdm or elmpt2cl 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.
• Exclusive or ( ph \/_ ps) is equivalent to ph <-> -. ps given excluded middle but we just have one direction (xorbin). Consider intuitionizing ph <-> -. ps as ph \/_ ps (example: rpnegap).
• 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
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
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
exmo	exmonim
mo2	mo2r, mo3
nne	nner, nnedc
exmidne	dcne
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
df-pss and all proper subclass theorems	none	In set.mm, "A is a proper subclass of B" is defined to be ( A C_ B /\ A =/= B ) and this definition is almost always used in conjunction with excluded middle. A more natural definition might be ( A C_ B /\ E. x x e. ( B \ A ) ), if we need proper subclass at all.
nss	nssr
ddif	ddifnel, ddifss
dfss4	ssddif, dfss4st
n0f	n0rf
n0	n0r, notm0, fin0, fin0or
neq0	neq0r
reximdva0	reximdva0m
ssdif0	ssdif0im
inssdif0	inssdif0im
abn0	abn0r, abn0m
rabn0	rabn0m, rabn0r
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
ssundif	ssundifim
uneqdifeq	uneqdifeqim
r19.2z	r19.2m
r19.9rzv	r19.9rmv
r19.45zv	r19.45mv
r19.44zv	r19.44mv
dfif2	df-if
ifsb	ifsbdc
dfif4	none	Unused in set.mm
dfif5	none	Unused in set.mm
ifeq1da	ifeq1dadc
ifnot	none
ifan	none
ifor	none
ifeq2da	none
ifclda	ifcldadc
ifeqda	none
elimif , ifid , ifval , elif , ifel , ifeqor , 2if2 , ifcomnan , csbif , csbifgOLD	none
ifbothda	ifbothdadc
ifboth	ifbothdc
eqif	eqifdc
ifcl , ifcld	ifcldcd, ifcldadc
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.
ifex , ifexg	ifcldcd, ifcldadc
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
sssn	sssnr, sssnm
pwsn	pwsnss	Also see exmidpw
iundif2	iundif2ss
iindif2	iindif2m
iinin2	iinin2m
iinin1	iinin1m
iinvdif	iindif2m	Unused in set.mm
riinn0	riinm
riinrab	iinrabm
iinuni	iinuniss
iununi	iununir
rintn0	rintm
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 e. _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 E! theorems still hold, but a decent number of the E* 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 } e. _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
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 om
	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
dmxpid	dmxpm
relimasn	imasng
rnxp	rnxpm
opswap	opswapg
cnvso	cnvsom
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	none	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
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 -. (/) e. { { (/) } } 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)
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 A =/= B 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
fvif	fvifdc
ndmfv	ndmfvg	The -. A e. _V case is fvprc.
elfvdm	relelfvdm
elfvex	relelfvdm
fvmpti	fvmptg	The set.mm proof relies on case elimination on C e. _V
fvmpt2i	fvmpt2	The set.mm proof relies on case elimination on C e. _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
fvunsn	fvunsng
fnsnsplit	none	Although it should be possible to prove subset rather than equality (using difsnss instead of difsnid ), this theorem is lightly used in set.mm.
funiunfv	fniunfv, funiunfvdm
funiunfvf	funiunfvdmf
eluniima	eluniimadm
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
nfriotad	nfriotadxy
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
	mpt2fvex	When the operation is defined via maps-to, yields a set on any inputs, and is being evaluated at two sets.
ovrcl	elmpt2cl, relelfvdm
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.
oprssdm	none
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	elmpt2cl, 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
ofval	fnofval
offn , offveq , 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. U. A = On -> A e/ V is provable, and ( B e. On /\ U. A C_ B ) -> A e. V is provable. (One thing we presumably could prove is ( U. A C_ On /\ E. x x e. ( On \ U. A ) ) -> A e. V 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 i^i 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 i^i 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 e. { (/) } | ph } 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	tfis3	We are unable to separate limit and successor ordinals using case elimination.
findsg	uzind4	findsg presumably could be proved, but there hasn't been a need for it.
xpexr2	xpexr2m
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.
brtpos	brtposg
ottpos	ottposg
ovtpos	ovtposg
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
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
xpider	xpiderm
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 .
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 { s | s C_ 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 , onfin2	none	The set.mm proofs rely on excluded middle
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 A ~<_ B /\ -. A ~~ B, 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 { (/) } e. om
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 e. Fin /\ B e. Fin /\ ( A u. B ) e. Fin ) -> ( A i^i B ) e. 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 , fimaxg	none	The set.mm proof 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.
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.
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 excluded middle.
fodomfi	none	Might be provable, for example via ac6sfi or induction directly. The set.mm proof does use undom in addition to induction.
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).
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.
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
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	Both the set.mm proof, and perhaps some possible alternative proofs, rely on onomeneq and perhaps other theorems we don't have currently.
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.
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 om ~<_ A follows definition 8.1.4 of [AczelRathjen], p. 71. It can presumably not be shown to be equivalent to -. A e. Fin in the absence of excluded middle (see inffiexmid which isn't exactly about -. A e. Fin <-> om ~<_ 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).
addcompi	addcompig
addasspi	addasspig
mulcompi	mulcompig
mulasspi	mulasspig
distrpi	distrpig
addcanpi	addcanpig
mulcanpi	mulcanpig
addnidpi	addnidpig
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.
addcomnq	addcomnqg
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
prpssnq	prssnql, prssnqu
elprnq	elprnql, elprnqu
prcdnq	prcdnql, prcunqu
prub	prubl
prnmax	prnmaxl
npomex	none
prnmadd	prnmaddl
genpv	genipv
genpcd	genpcdl
genpnmax	genprndl
ltrnq	ltrnqg, ltrnqi
genpcl	addclpr, mulclpr
genpass	genpassg
addclprlem1	addnqprllem, addnqprulem
addclprlem2	addnqprl, addnqpru
plpv	plpvlu
mpv	mpvlu
nqpr	nqprlu
mulclprlem	mulnqprl, mulnqpru
addcompr	addcomprg
addasspr	addassprg
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
ltaddpr2	ltaddpr
ltexprlem1 , ltexprlem2 , ltexprlem3 , ltexprlem4	none	See the lemmas for ltexpri generally.
ltexprlem5	ltexprlempr
ltexprlem6	ltexprlemfl, ltexprlemfu
ltexprlem7	ltexprlemrl, ltexprlemru
ltapr	ltaprg
addcanpr	addcanprg
prlem936	prmuloc2
reclem2pr	recexprlempr
reclem3pr	recexprlem1ssl, recexprlem1ssu
reclem4pr	recexprlemss1l, recexprlemss1u, recexprlemex
supexpr , suplem1pr , suplem2pr	caucvgprpr	The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle. We express completeness using sequences.
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 X. T ) /. R )), we haven't yet had a need to do so.
addcomsr	addcomsrg
addasssr	addasssrg
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 , ltpsrpr , map2psrpr	prsrpos, prsrlt, srpospr
supsrlem , supsr	caucvgsr	The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle. We express completeness using sequences.
axaddf , ax-addf , axmulf , ax-mulf	none	Because these are described as deprecated in set.mm, we haven't figured out what would be involved in proving them for iset.mm.
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 , axsup	axcaucvg, ax-caucvg, caucvgre	The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle. We express completeness using sequences.
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	none
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.
ltordlem , ltord1 , leord1 , eqord1 , ltord2 , leord2 , eqord2	none	Although these presumably could be proved using theorems like letri3 and lenlt, at least for now we have chosen to just invoke those other theorems directly (example: expcan and its lemma expcanlem) which avoids some extra set variables and produces proofs which are almost as short.
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.
mulcanad , mulcan2ad	mulcanapad, mulcanap2ad	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	none	Remark 2.19 of [Geuvers] says that this does not hold in general and has a counterexample.
mulne0b , mulne0bd , mulne0bad , mulne0bbd	mulap0b, mulap0bd, mulap0bad, mulap0bbd
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
ddcan , divadddiv , divsubdiv	ddcanap, divadddivap, divsubdivap
ddcan , divadddiv , divsubdiv	ddcanap, divadddivap, divsubdivap
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
divmuldivi, divmul13i, divadddivi, divdivdivi	divmuldivapi, divmul13api, divadddivapi, divdivdivapi
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
mvllmuld	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	fimaxre2	When applied to a pair this 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 , sup3 , sup3ii	none	We won't be able to have the least upper bound property for all nonempty bounded sets. In cases where we can show that the supremum exists, we might be able to prove slightly different ways of stating there is a supremum.
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
halfthird	none	Presumably will be easy to intuitionize
5recm6rec	none	Presumably will be easy to intuitionize
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
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 , xrmax2 , xrmin1 , xrmin2 , xrmaxeq , xrmineq , xrmaxlt , xrltmin , xrmaxle , xrlemin , max0sub , ifle	none
max1	maxle1
max2	maxle2
2resupmax	none	Proved from real trichotomy. We express maximum in iset.mm using sup ( { A , B } , RR , < ) rather than if ( A <_ B , B , A ). The former has the expected maximum properties such as maxcl, maxle1, maxle2, maxleast, and maxleb.
min1	min1inf
min2	min2inf
maxle	maxleastb
lemin	lemininf
maxlt	maxltsup
ltmin	ltmininf
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
xaddval , xaddf , xmulval , xaddpnf1 , xaddpnf2 , xaddmnf1 , xaddmnf2 , pnfaddmnf , mnfaddpnf , rexadd , rexsub , xaddnemnf , xaddnepnf , xnegid , xaddcl , xaddcom , xaddid1 , xaddid2 , xnegdi , xaddass , xaddass2 , xpncan , xnpcan , xleadd1a , xleadd2a , xleadd1 , xltadd1 , xltadd2 , xaddge0 , xle2add , xlt2add , xsubge0 , xposdif , xlesubadd , 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 , xaddcld , xmulcld , xadd4d	none	There appears to be no fundamental obstacle to proving these, because disjunctions can arise from elxr rather than excluded middle. However, -e, +e, and xe are lightly used in set.mm, and the set.mm proofs would require significant changes.
ixxub , ixxlb	none
iccen	none
supicc , supiccub , supicclub , supicclub2	none
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	none
icc0	icc0r
ioorebas	ioorebasg
ge0xaddcl , ge0xmulcl	none	Rely on xaddcl and xmulcl ; see discussion in this list for those 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 e. _V and N e. _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
fladdz	flqaddz
flzadd	flqzadd
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
modadd1	modqadd1
modaddabs	modqaddabs
modaddmod	modqaddmod
muladdmodid	mulqaddmodid
modmuladd	modqmuladd
modmuladdim	modqmuladdim
modmuladdnn0	modqmuladdnn0
negmod	qnegmod
modadd2mod	modqadd2mod
modm1p1mod0	modqm1p1mod0
modltm1p1mod	modqltm1p1mod
modmul1	modqmul1
modmul12d	modqmul12d
modnegd	modqnegd
modadd12d	modqadd12d
modsub12d	modqsub12d
modsubmod	modqsubmod
modsubmodmod	modqsubmodmod
2txmodxeq0	q2txmodxeq0
2submod	q2submod
modmulmod	modqmulmod
modmulmodr	modqmulmodr
modaddmulmod	modqaddmulmod
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-iseq
seqex	iseqex	The only difference is the differing syntax of seq.
seqeq1 , seqeq2 , seqeq3 , seqeq1d , seqeq2d , seqeq3d , seqeq123d	iseqeq1, iseqeq2, iseqeq3
nfseq	nfiseq
seqval	iseqval
seqfn	iseqfcl
seq1 , seq1i	iseq1
seqp1 , seqp1i	iseqp1
seqm1	iseqm1
seqcl2	none yet	Presumably could prove this (with adjustments analogous to iseqp1). The third argument to seq in iset.mm would correspond to C rather than D (some of the other seq related theorems do not allow C and D to be different, but seqcl2 does).
seqf2	none yet	Presumably could prove this, analogously to seqcl2.
seqcl	iseqcl	iseqcl requires that F be defined on ( ZZ>= ` M ) not merely ( M ... N ) as in seqcl . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqf	iseqfcl
seqfveq2	iseqfveq2
seqfeq2	iseqfeq2
seqfveq	iseqfveq
seqfeq	iseqfeq
seqshft2	iseqshft2
seqres	none	Should be intuitionizable as with the other seq theorems, but unused in set.mm
serf	iserf
serfre	iserfre
sermono	isermono	isermono requires that F be defined on ( ZZ>= ` M ) not merely ( M ... N ) as in sermono . Given that the intention is to use this for infinite series, there may be no need to look into whether this requirement can be relaxed.
seqsplit	iseqsplit	iseqsplit requires that F be defined on ( ZZ>= ` K ) not merely ( K ... N ) as in seqsplit . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seq1p	iseq1p	iseq1p requires that F be defined on ( ZZ>= ` M ) not merely ( M ... N ) as in seq1p . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqcaopr3	iseqcaopr3	iseqcaopr3 requires that F, G, and H be defined on ( ZZ>= ` M ) not merely ( M ... N ) as in seqcaopr3 . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqcaopr2	iseqcaopr2	iseqcaopr2 requires that F, G, and H be defined on ( ZZ>= ` M ) not merely ( M ... N ) as in seqcaopr2 . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqcaopr	iseqcaopr	iseqcaopr requires that F, G, and H be defined on ( ZZ>= ` M ) not merely ( M ... N ) as in seqcaopr . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqf1o	none	May be adaptable with sufficient attention to issues such as whether the operation .+ is closed with respect to C as well as S, and perhaps other issues.
seradd	iseradd	iseradd requires that F, G, and H be defined on ( ZZ>= ` M ) not merely ( M ... N ) as in seradd . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
sersub	isersub	isersub requires that F, G, and H be defined on ( ZZ>= ` M ) not merely ( M ... N ) as in sersub . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqid3	iseqid3, iseqid3s	iseqid3 and iseqid3s differ with respect to which entries in F need to be zero. Further refinements to the hypotheses including where F needs to be defined and the like might be possible (with a greater amount of work), but perhaps these are sufficient.
seqid	iseqid
seqid2	iseqid2
seqhomo	iseqhomo
seqz	iseqz
seqfeq4 , seqfeq3	none yet	It should be possible to come up with some (presumably modified) versions of these, but we have not done so yet.
seqdistr	iseqdistr
ser0	iser0	The only difference is the syntax of seq.
ser0f	iser0f	The only difference is the syntax of seq.
serge0	serige0	This theorem uses CC as the final argument to seq which probably will be more convenient even if all the values are elements of a subset of CC.
serle	serile	This theorem uses CC as the final argument to seq which probably will be more convenient even if all the values are elements of a subset of CC.
ser1const , seqof , seqof2	none yet	It should be possible to come up with some (presumably modified) versions of these, but we have not done so yet.
df-exp	df-iexp	The difference is that seq in iset.mm takes three arguments compared with two in set.mm.
expval	expival
expnnval	expinnval
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
expaddzlem , expaddz	expaddzaplem, expaddzap
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.
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.
bcval5	ibcval5
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 e. Fin to om ~<_ 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
hashen1	fihashen1
hashrabrsn	none	Presumably would need conditions around the existence of A and decidability of ph but unused in set.mm.
hashrabsn01	none	Presumably would need conditions around the existence of A and decidability of ph 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.
hashgadd	omgadd
hashgval2	none	Presumably provable, when restated as (♯ |` om ) = frec ( ( x e. ZZ |-> ( 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
hashinfxadd	none
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 e. _V or -. N e. _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
prsshashgt1	fiprsshashgt1
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
hashunlei	none	Not provable per unfiexmid (see also entry for hashun2)
hashsslei	fihashss	Would be provable if we transfered B e. 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 e. Fin or similar conditions.
hashimarn , hashimarni	none	Presumably would need an additional condition such as F e. Fin but unused in set.mm.
fnfz0hashnn0	none	Presumably would need an additional condition such as N e. NN0 but unused in set.mm.
fnfzo0hashnn0	none	Presumably would need an additional condition such as N e. NN0 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 , leisorel	none	The set.mm proof uses isocnv3 .
fz1iso	none	The set.mm proof depends on OrdIso
ishashinf	none	May be possible with the antecedent changed from -. A e. Fin to om ~<_ A but the set.mm proof does not work as is.
seqcoll , seqcoll2	none	Presumably will require significant modifications.
seqshft	none currently	need to figure out how to adjust this for the differences between set.mm and iset.mm concerning seq.
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 ( Im ` x ) # 0 \/ 0 <_ ( Re ` 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
max0add	max0addsup
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.
serclim0	iserclim0	The only difference is the syntax of seq.
reccn2	none yet	Will need to be revamped to deal with negated equality versus apartness and perhaps other issues.
clim2ser	clim2iser	The only difference is the syntax of seq.
clim2ser2	clim2iser2	The only difference is the syntax of seq.
iserex	iiserex	The only difference is the syntax of seq.
isermulc2	iisermulc2	The only difference is the syntax of seq.
iserle	iserile	The only difference is the syntax of seq.
iserge0	iserige0	The only difference is the syntax of seq.
climserle	climserile	The only difference is the syntax of seq.
isershft	none yet	Relies on seqshft
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.
serf0	serif0	The only difference is the syntax of seq.
iseralt	none	The set.mm proof relies on caurcvg2 which does not specify a rate of convergence.
df-sum	df-isum	Although this defintion is intended to function similarly to the set.mm one, a lot of details have to be changed, especially around decidability, to make sum work.
sumex	none	This will need to replaced by suitable closure theorems.
sumeq2w	sumeq2	Presumably could be proved, and perhaps also would rely only on extensionality (and logical axioms). But unused in set.mm.
sumeq2ii	sumeq2d
sumrblem	isumrblem
fsumcvg	fisumcvg
sumrb	isumrb
df-prod and theorems using it	none	To define this, will need to tackle all the issues with df-isum plus some more around, for example, not equal to zero versus apart from zero
znnen	none	Corollary 8.1.23 of [AczelRathjen] and thus presumably provable. The set.mm proof would not work as-is or with small changes, however.
qnnen	none	Corollary 8.1.23 of [AczelRathjen] and thus presumably provable. The set.mm proof would not work as-is or with small changes, however.
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 second argument to seq, at least as handled in theorems such as iseqfcl, 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	Several hypotheses are tweaked or added to reflect differences in how seq works
algrf	ialgrf	Several hypotheses are tweaked or added to reflect differences in how seq works
algrp1	ialgrp1	Several hypotheses are tweaked or added to reflect differences in how seq works
alginv	ialginv	Two hypotheses are tweaked or added to reflect differences in how seq works
algcvg	ialgcvg	seq as defined in df-iseq has a third argument
algcvga	ialgcvga	Two hypotheses are tweaked or added to reflect differences in how seq works
algfx	ialgfx	Two hypotheses are tweaked or added to reflect differences in how seq works
eucalg	eucialg	Tweaked for the different seq syntax
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

Bibliography   

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