P. 165 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

eupth2lembfi 16401*

Lemma for eupth2fi 16403 (induction basis): There are no vertices of odd degree in an Eulerian path of length 0, having no edge and identical endpoints (the single vertex of the Eulerian path). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = (/) )

Theorem

eupth2lemsfi 16402*

Lemma for eupth2fi 16403 (induction step): The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct, if the Eulerian path shortened by one edge has this property. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> V e. Fin )   =>    |- ( ( ph /\ n e. NN0 ) -> ( ( n <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) -> ( ( n + 1 ) <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) ) )

Theorem

eupth2fi 16403*

The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> Fun I )   &    |- ( ph -> F (EulerPaths ` G ) P )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) )

Theorem

eulerpathprum 16404*

A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   =>    |- ( ( G e. UMGraph /\ F (EulerPaths ` G ) P /\ V e. Fin ) -> (♯ ` { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } ) e. { 0 , 2 } )

Theorem

eulerpathum 16405*

A multigraph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)

|- V = (Vtx ` G )   =>    |- ( ( G e. UMGraph /\ E. j j e. (EulerPaths ` G ) /\ V e. Fin ) -> (♯ ` { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } ) e. { 0 , 2 } )

12.4.2  The Königsberg Bridge problem

According to Wikipedia ("Seven Bridges of Königsberg", 9-Mar-2021, https://en.wikipedia.org/wiki/Seven_Bridges_of_Koenigsberg): "The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology. The city of Königsberg in [East] Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands - Kneiphof and Lomse - which were connected to each other, or to the two mainland portions of the city, by seven bridges. The problem was to devise a walk through the city that would cross each of those bridges once and only once.". Euler proved that the problem has no solution by applying Euler's theorem to the Königsberg graph, which is obtained by replacing each land mass with an abstract "vertex" or node, and each bridge with an abstract connection, an "edge", which connects two land masses/vertices. The Königsberg graph G is a multigraph consisting of 4 vertices and 7 edges, represented by the following ordered pair: G = <. ( 0 ... 3 ) , <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } "> >., see konigsbergumgr 16411. konigsberg 16417 shows that the Königsberg graph has no Eulerian path, thus the Königsberg Bridge problem has no solution.

Theorem

konigsbergvtx 16406

The set of vertices of the Königsberg graph G. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- (Vtx ` G ) = ( 0 ... 3 )

Theorem

konigsbergiedg 16407

The indexed edges of the Königsberg graph G. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- (iEdg ` G ) = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">

Theorem

konigsbergiedgwen 16408*

The indexed edges of the Königsberg graph G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- E e. Word { x e. ~P V | x ~~ 2o }

Theorem

konigsbergssiedgwpren 16409*

Each subset of the indexed edges of the Königsberg graph G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( ( A e. Word _V /\ B e. Word _V /\ E = ( A ++ B ) ) -> A e. Word { x e. ~P V | x ~~ 2o } )

Theorem

konigsbergssiedgwen 16410*

Each subset of the indexed edges of the Königsberg graph G is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( ( A e. Word _V /\ B e. Word _V /\ E = ( A ++ B ) ) -> A e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )

Theorem

konigsbergumgr 16411

The Königsberg graph G is a multigraph. (Contributed by AV, 28-Feb-2021.) (Revised by AV, 9-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- G e. UMGraph

Theorem

konigsberglem1 16412

Lemma 1 for konigsberg 16417: Vertex 0 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( (VtxDeg ` G ) ` 0 ) = 3

Theorem

konigsberglem2 16413

Lemma 2 for konigsberg 16417: Vertex 1 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( (VtxDeg ` G ) ` 1 ) = 3

Theorem

konigsberglem3 16414

Lemma 3 for konigsberg 16417: Vertex 3 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- ( (VtxDeg ` G ) ` 3 ) = 3

Theorem

konigsberglem4 16415*

Lemma 4 for konigsberg 16417: Vertices 0 , 1 , 3 are vertices of odd degree. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- { 0 , 1 , 3 } C_ { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) }

Theorem

konigsberglem5 16416*

Lemma 5 for konigsberg 16417: The set of vertices of odd degree is greater than 2. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- 2 < (♯ ` { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } )

Theorem

konigsberg 16417

The Königsberg Bridge problem. If G is the Königsberg graph, i.e. a graph on four vertices 0 , 1 , 2 , 3, with edges { 0 , 1 } , { 0 , 2 } , { 0 , 3 } , { 1 , 2 } , { 1 , 2 } , { 2 , 3 } , { 2 , 3 }, then vertices 0 , 1 , 3 each have degree three, and 2 has degree five, so there are four vertices of odd degree and thus by eulerpathum 16405 the graph cannot have an Eulerian path. It is sufficient to show that there are 3 vertices of odd degree, since a graph having an Eulerian path can only have 0 or 2 vertices of odd degree. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 9-Mar-2021.)

|- V = ( 0 ... 3 )   &    |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">   &    |- G = <. V , E >.   =>    |- (EulerPaths ` G ) = (/)

PART 13  GUIDES AND MISCELLANEA

13.1  Guides (conventions, explanations, and examples)

13.1.1  Conventions

This section describes the conventions we use. These conventions often refer to existing mathematical practices, which are discussed in more detail in other references. The following sources lay out how mathematics is developed without the law of the excluded middle. Of course, there are a greater number of sources which assume excluded middle and most of what is in them applies here too (especially in a treatment such as ours which is built on first-order logic and set theory, rather than, say, type theory). Studying how a topic is treated in the Metamath Proof Explorer and the references therein is often a good place to start (and is easy to compare with the Intuitionistic Logic Explorer). The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

• Axioms of propositional calculus - Stanford Encyclopedia of Philosophy or [Heyting].
• Axioms of predicate calculus - our axioms are adapted from the ones in the Metamath Proof Explorer.
• Theorems of propositional calculus - [Heyting].
• Theorems of pure predicate calculus - Metamath Proof Explorer.
• Theorems of equality and substitution - Metamath Proof Explorer.
• Axioms of set theory - [Crosilla].
• Development of set theory - Chapter 10 of [HoTT].
• Construction of real and complex numbers - Chapter 11 of [HoTT]; [BauerTaylor].
• Theorems about real numbers - [Geuvers].

Theorem

conventions 16418

Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (MPE, set.mm). This list of conventions is intended to be read in conjunction with the corresponding conventions in the Metamath Proof Explorer, and only the differences are described below.

• Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, our choice of IZF (Intuitionistic Zermelo-Fraenkel) over CZF (Constructive Zermelo-Fraenkel, a weaker system) was just an expedient choice because IZF is easier to formalize in Metamath. You can find some development using CZF in BJ's mathbox starting at wbd 16511 (and the section header just above it). As for the axiom of choice, the full axiom of choice implies excluded middle as seen at acexmid 6027, although some authors will use countable choice or dependent choice. For example, countable choice or excluded middle is needed to show that the Cauchy reals coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT], p. (varies).
• Junk/undefined results. Much of the discussion of this topic in the Metamath Proof Explorer applies except that certain techniques are not available to us. For example, the Metamath Proof Explorer will often say "if a function is evaluated within its domain, a certain result follows; if the function is evaluated outside its domain, the same result follows. Since the function must be evaluated within its domain or outside it, the result follows unconditionally" (the use of excluded middle in this argument is perhaps obvious when stated this way). Often, the easiest fix will be to prove we are evaluating functions within their domains, other times it will be possible to use a theorem like relelfvdm 5680 which says that if a function value produces an inhabited set, then the function is being evaluated within its domain.
• Bibliography references. The bibliography for the Intuitionistic Logic Explorer is separate from the one for the Metamath Proof Explorer but feel free to copy-paste a citation in either direction in order to cite it.

Label naming conventions

Here are a few of the label naming conventions:

• Suffixes. We follow the conventions of the Metamath Proof Explorer with a few additions. A biconditional in set.mm which is an implication in iset.mm should have a "r" (for the reverse direction), or "i"/"im" (for the forward direction) appended. A theorem in set.mm which has a decidability condition added should add "dc" to the theorem name. A theorem in set.mm where "nonempty class" is changed to "inhabited class" should add "m" (for member) to the theorem name.
• iset.mm versus set.mm names

  Theorems which are the same as in set.mm should be named the same (that is, where the statement of the theorem is the same; the proof can differ without a new name being called for). Theorems which are different should be named differently (we do have a small number of intentional exceptions to this rule but on the whole it serves us well).

  As for how to choose names so they are different between iset.mm and set.mm, when possible choose a name which reflect the difference in the theorems. For example, if a theorem in set.mm is an equality and the iset.mm analogue is a subset, add "ss" to the iset.mm name. If need be, add "i" to the iset.mm name (usually as a prefix to some portion of the name).

  As with set.mm, we welcome suggestions for better names (such as names which are more consistent with naming conventions).

  We do try to keep set.mm and iset.mm similar where we can. For example, if a theorem exists in both places but the name in set.mm isn't great, we tend to keep that name for iset.mm, or change it in both files together. This is mainly to make it easier to copy theorems, but also to generally help people browse proofs, find theorems, write proofs, etc.

The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.

For the "g" abbreviation, this is related to the set.mm usage, in which "is a set" conditions are converted from hypotheses to antecedents, but is also used where "is a set" conditions are added relative to similar set.mm theorems.

Abbreviation	Mnenomic/Meaning	Source	Expression	Syntax?	Example(s)
ap	apart	df-pap 7510, df-ap 8804		Yes	apadd1 8830, apne 8845
g	with "is a set" condition			No	1stvalg 6314, brtposg 6463, setsmsbasg 15273
m	inhabited (from "member")		E. x x e. A	No	r19.2m 3583, negm 9893, ctm 7351, basmex 13205
seq3, sum3	recursive sequence	df-seqfrec 10756		Yes	seq3-1 10770, fsum3 12011
tap	tight apartness	df-tap 7512		Yes	df-tap 7512

• Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7512.

(Contributed by Jim Kingdon, 24-Feb-2020.) (New usage is discouraged.)

|- ph   =>    |- ph

13.1.2  Definitional examples

Theorem

ex-or 16419

Example for ax-io 717. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

|- ( 2 = 3 \/ 4 = 4 )

Theorem

ex-an 16420

Example for ax-ia1 106. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

|- ( 2 = 2 /\ 3 = 3 )

Theorem

1kp2ke3k 16421

Example for df-dec 9656, 1000 + 2000 = 3000.

This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with ( 2 + 1 ) = 3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 9656 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

|- (;;; 1 0 0 0 + ;;; 2 0 0 0 ) = ;;; 3 0 0 0

Theorem

ex-fl 16422

Example for df-fl 10576. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

|- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 )

Theorem

ex-ceil 16423

Example for df-ceil 10577. (Contributed by AV, 4-Sep-2021.)

|- ( (⌈ ` ( 3 / 2 ) ) = 2 /\ (⌈ ` -u ( 3 / 2 ) ) = -u 1 )

Theorem

ex-exp 16424

Example for df-exp 10847. (Contributed by AV, 4-Sep-2021.)

|- ( ( 5 ^ 2 ) = ; 2 5 /\ ( -u 3 ^ -u 2 ) = ( 1 / 9 ) )

Theorem

ex-fac 16425

Example for df-fac 11034. (Contributed by AV, 4-Sep-2021.)

|- ( ! ` 5 ) = ;; 1 2 0

Theorem

ex-bc 16426

Example for df-bc 11056. (Contributed by AV, 4-Sep-2021.)

|- ( 5 _C 3 ) = ; 1 0

Theorem

ex-dvds 16427

Example for df-dvds 12412: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

|- 3 || 6

Theorem

ex-gcd 16428

Example for df-gcd 12588. (Contributed by AV, 5-Sep-2021.)

|- ( -u 6 gcd 9 ) = 3

PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

14.1  Mathboxes for user contributions

14.1.1  Mathbox guidelines

Theorem

mathbox 16429

(This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm.

Guidelines:

Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details.

(Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.)

|- ph   =>    |- ph

14.2  Mathbox for Matthew House

Theorem

depindlem1 16430*

Lemma for depind 16433. (Contributed by Matthew House, 14-Apr-2026.)

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> ( F : NN0 --> _V /\ ( F ` 0 ) = A /\ A. n e. NN0 ( F ` ( n + 1 ) ) = ( ( H ` n ) ` ( F ` n ) ) ) )

Theorem

depindlem2 16431*

Lemma for depind 16433. (Contributed by Matthew House, 14-Apr-2026.)

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> F e. X_ n e. NN0 ( P ` n ) )

Theorem

depindlem3 16432*

Lemma for depind 16433. (Contributed by Matthew House, 14-Apr-2026.)

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> A. f e. X_ n e. NN0 ( P ` n ) ( ( ( f ` 0 ) = A /\ A. n e. NN0 ( f ` ( n + 1 ) ) = ( ( H ` n ) ` ( f ` n ) ) ) -> f = F ) )

Theorem

depind 16433*

Theorem related to a dependently typed induction principle in type theory. (Contributed by Matthew House, 14-Apr-2026.)

|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   =>    |- ( ph -> E! f e. X_ n e. NN0 ( P ` n ) ( ( f ` 0 ) = A /\ A. n e. NN0 ( f ` ( n + 1 ) ) = ( ( H ` n ) ` ( f ` n ) ) ) )

14.3  Mathbox for BJ

14.3.1  Propositional calculus

Theorem

bj-nnsn 16434

As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)

|- ( ( ph -> -. ps ) <-> ( -. -. ph -> -. ps ) )

Theorem

bj-nnor 16435

Double negation of a disjunction in terms of implication. (Contributed by BJ, 9-Oct-2019.)

|- ( -. -. ( ph \/ ps ) <-> ( -. ph -> -. -. ps ) )

Theorem

bj-nnim 16436

The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)

|- ( -. -. ( ph -> ps ) -> ( ph -> -. -. ps ) )

Theorem

bj-nnan 16437

The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)

|- ( -. -. ( ph /\ ps ) -> ( -. -. ph /\ -. -. ps ) )

Theorem

bj-nnclavius 16438

Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)

|- ( ( -. ph -> ph ) -> -. -. ph )

Theorem

bj-imnimnn 16439

If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 16438 as its last step. (Contributed by BJ, 27-Oct-2024.)

|- ( ph -> ps )   &    |- ( -. ph -> ps )   =>    |- -. -. ps

14.3.1.1  Stable formulas

Some of the following theorems, like bj-sttru 16441 or bj-stfal 16443 could be deduced from their analogues for decidability, but stability is not provable from decidability in minimal calculus, so direct proofs have their interest.

Theorem

bj-trst 16440

A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)

|- ( ph -> STAB ph )

Theorem

bj-sttru 16441

The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)

|- STAB T.

Theorem

bj-fast 16442

A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)

|- ( -. ph -> STAB ph )

Theorem

bj-stfal 16443

The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)

|- STAB F.

Theorem

bj-nnst 16444

Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 16689 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( -> , -. ) -intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in ( -> , <-> , -. )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)

|- -. -. STAB ph

Theorem

bj-nnbist 16445

If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if ph is a classical tautology, then -. -. ph is an intuitionistic tautology. Therefore, if ph is a classical tautology, then ph is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 16458). (Contributed by BJ, 24-Nov-2023.)

|- ( -. -. ph -> (STAB ph <-> ph ) )

Theorem

bj-stst 16446

Stability of a proposition is stable if and only if that proposition is stable. STAB is idempotent. (Contributed by BJ, 9-Oct-2019.)

|- (STAB STAB ph <-> STAB ph )

Theorem

bj-stim 16447

A conjunction with a stable consequent is stable. See stabnot 841 for negation , bj-stan 16448 for conjunction , and bj-stal 16450 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

|- (STAB ps -> STAB ( ph -> ps ) )

Theorem

bj-stan 16448

The conjunction of two stable formulas is stable. See bj-stim 16447 for implication, stabnot 841 for negation, and bj-stal 16450 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

|- ( (STAB ph /\ STAB ps ) -> STAB ( ph /\ ps ) )

Theorem

bj-stand 16449

The conjunction of two stable formulas is stable. Deduction form of bj-stan 16448. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 16448 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

|- ( ph -> STAB ps )   &    |- ( ph -> STAB ch )   =>    |- ( ph -> STAB ( ps /\ ch ) )

Theorem

bj-stal 16450

The universal quantification of a stable formula is stable. See bj-stim 16447 for implication, stabnot 841 for negation, and bj-stan 16448 for conjunction. (Contributed by BJ, 24-Nov-2023.)

|- ( A. xSTAB ph -> STAB A. x ph )

Theorem

bj-pm2.18st 16451

Clavius law for stable formulas. See pm2.18dc 863. (Contributed by BJ, 4-Dec-2023.)

|- (STAB ph -> ( ( -. ph -> ph ) -> ph ) )

Theorem

bj-con1st 16452

Contraposition when the antecedent is a negated stable proposition. See con1dc 864. (Contributed by BJ, 11-Nov-2024.)

|- (STAB ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )

14.3.1.2  Decidable formulas

Theorem

bj-trdc 16453

A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)

|- ( ph -> DECID ph )

Theorem

bj-dctru 16454

The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)

|- DECID T.

Theorem

bj-fadc 16455

A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)

|- ( -. ph -> DECID ph )

Theorem

bj-dcfal 16456

The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)

|- DECID F.

Theorem

bj-dcstab 16457

A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

|- (DECID ph -> STAB ph )

Theorem

bj-nnbidc 16458

If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 16445. (Contributed by BJ, 24-Nov-2023.)

|- ( -. -. ph -> (DECID ph <-> ph ) )

Theorem

bj-nndcALT 16459

Alternate proof of nndc 859. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)

|- -. -. DECID ph

Theorem

bj-dcdc 16460

Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)

|- (DECID DECID ph <-> DECID ph )

Theorem

bj-stdc 16461

Decidability of a proposition is stable if and only if that proposition is decidable. In particular, the assumption that every formula is stable implies that every formula is decidable, hence classical logic. (Contributed by BJ, 9-Oct-2019.)

|- (STAB DECID ph <-> DECID ph )

Theorem

bj-dcst 16462

Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)

|- (DECID STAB ph <-> STAB ph )

14.3.2  Predicate calculus

Theorem

bj-ex 16463*

Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1647 and 19.9ht 1690 or 19.23ht 1546). (Proof modification is discouraged.)

|- ( E. x ph -> ph )

Theorem

bj-hbalt 16464

Closed form of hbal 1526 (copied from set.mm). (Contributed by BJ, 2-May-2019.)

|- ( A. y ( ph -> A. x ph ) -> ( A. y ph -> A. x A. y ph ) )

Theorem

bj-nfalt 16465

Closed form of nfal 1625 (copied from set.mm). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)

|- ( A. x F/ y ph -> F/ y A. x ph )

Theorem

spimd 16466

Deduction form of spim 1786. (Contributed by BJ, 17-Oct-2019.)

|- ( ph -> F/ x ch )   &    |- ( ph -> A. x ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( A. x ps -> ch ) )

Theorem

2spim 16467*

Double substitution, as in spim 1786. (Contributed by BJ, 17-Oct-2019.)

|- F/ x ch   &    |- F/ z ch   &    |- ( ( x = y /\ z = t ) -> ( ps -> ch ) )   =>    |- ( A. z A. x ps -> ch )

Theorem

ch2var 16468*

Implicit substitution of y for x and t for z into a theorem. (Contributed by BJ, 17-Oct-2019.)

|- F/ x ps   &    |- F/ z ps   &    |- ( ( x = y /\ z = t ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps

Theorem

ch2varv 16469*

Version of ch2var 16468 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)

|- ( ( x = y /\ z = t ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps

Theorem

bj-exlimmp 16470

Lemma for bj-vtoclgf 16477. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

|- F/ x ps   &    |- ( ch -> ph )   =>    |- ( A. x ( ch -> ( ph -> ps ) ) -> ( E. x ch -> ps ) )

Theorem

bj-exlimmpi 16471

Lemma for bj-vtoclgf 16477. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

|- F/ x ps   &    |- ( ch -> ph )   &    |- ( ch -> ( ph -> ps ) )   =>    |- ( E. x ch -> ps )

Theorem

bj-sbimedh 16472

A strengthening of sbiedh 1835 (same proof). (Contributed by BJ, 16-Dec-2019.)

|- ( ph -> A. x ph )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( [ y / x ] ps -> ch ) )

Theorem

bj-sbimeh 16473

A strengthening of sbieh 1838 (same proof). (Contributed by BJ, 16-Dec-2019.)

|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( [ y / x ] ph -> ps )

Theorem

bj-sbime 16474

A strengthening of sbie 1839 (same proof). (Contributed by BJ, 16-Dec-2019.)

|- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( [ y / x ] ph -> ps )

14.3.3  Set theorey miscellaneous

Theorem

bj-el2oss1o 16475

Shorter proof of el2oss1o 6654 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( A e. 2o -> A C_ 1o )

14.3.4  Extensionality

Various utility theorems using FOL and extensionality.

Theorem

bj-vtoclgft 16476

Weakening two hypotheses of vtoclgf 2863. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ph )   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. V -> ps ) )

Theorem

bj-vtoclgf 16477

Weakening two hypotheses of vtoclgf 2863. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ph )   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. V -> ps )

Theorem

elabgf0 16478

Lemma for elabgf 2949. (Contributed by BJ, 21-Nov-2019.)

|- ( x = A -> ( A e. { x | ph } <-> ph ) )

Theorem

elabgft1 16479

One implication of elabgf 2949, in closed form. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) )

Theorem

elabgf1 16480

One implication of elabgf 2949. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )

Theorem

elabgf2 16481

One implication of elabgf 2949. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. B -> ( ps -> A e. { x | ph } ) )

Theorem

elabf1 16482*

One implication of elabf 2950. (Contributed by BJ, 21-Nov-2019.)

|- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )

Theorem

elabf2 16483*

One implication of elabf 2950. (Contributed by BJ, 21-Nov-2019.)

|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( ps -> A e. { x | ph } )

Theorem

elab1 16484*

One implication of elab 2951. (Contributed by BJ, 21-Nov-2019.)

|- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )

Theorem

elab2a 16485*

One implication of elab 2951. (Contributed by BJ, 21-Nov-2019.)

|- A e. _V   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( ps -> A e. { x | ph } )

Theorem

elabg2 16486*

One implication of elabg 2953. (Contributed by BJ, 21-Nov-2019.)

|- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. V -> ( ps -> A e. { x | ph } ) )

Theorem

bj-rspgt 16487

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2908 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/_ x B   &    |- F/ x ps   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A. x e. B ph -> ( A e. B -> ps ) ) )

Theorem

bj-rspg 16488

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2908 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

|- F/_ x A   &    |- F/_ x B   &    |- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A. x e. B ph -> ( A e. B -> ps ) )

Theorem

cbvrald 16489*

Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)

|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) )

Theorem

bj-intabssel 16490

Version of intss1 3948 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

|- F/_ x A   =>    |- ( A e. V -> ( [. A / x ]. ph -> |^| { x | ph } C_ A ) )

Theorem

bj-intabssel1 16491

Version of intss1 3948 using a class abstraction and implicit substitution. Closed form of intmin3 3960. (Contributed by BJ, 29-Nov-2019.)

|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. V -> ( ps -> |^| { x | ph } C_ A ) )

Theorem

bj-elssuniab 16492

Version of elssuni 3926 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

|- F/_ x A   =>    |- ( A e. V -> ( [. A / x ]. ph -> A C_ U. { x | ph } ) )

Theorem

bj-sseq 16493

If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.)

|- ( ph -> ( ps <-> A C_ B ) )   &    |- ( ph -> ( ch <-> B C_ A ) )   =>    |- ( ph -> ( ( ps /\ ch ) <-> A = B ) )

14.3.5  Decidability of classes

The question of decidability is essential in intuitionistic logic. In intuitionistic set theories, it is natural to define decidability of a set (or class) as decidability of membership in it. One can parameterize this notion with another set (or class) since it is often important to assess decidability of membership in one class among elements of another class. Namely, one will say that " A is decidable in B " if A. x e. BDECID x e. A (see df-dcin 16495).

Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 16540).

Syntax

wdcin 16494

Syntax for decidability of a class in another.

wff A DECID_in B

Definition

df-dcin 16495*

Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)

|- ( A DECID_in B <-> A. x e. B DECID x e. A )

Theorem

decidi 16496

Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)

|- ( A DECID_in B -> ( X e. B -> ( X e. A \/ -. X e. A ) ) )

Theorem

decidr 16497*

Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)

|- ( ph -> ( x e. B -> ( x e. A \/ -. x e. A ) ) )   =>    |- ( ph -> A DECID_in B )

Theorem

decidin 16498

If A is a decidable subclass of B (meaning: it is a subclass of B and it is decidable in B), and B is decidable in C, then A is decidable in C. (Contributed by BJ, 19-Feb-2022.)

|- ( ph -> A C_ B )   &    |- ( ph -> A DECID_in B )   &    |- ( ph -> B DECID_in C )   =>    |- ( ph -> A DECID_in C )

Theorem

uzdcinzz 16499

An upperset of integers is decidable in the integers. Reformulation of eluzdc 9888. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)

|- ( M e. ZZ -> ( ZZ>= ` M ) DECID_in ZZ )

Theorem

sumdc2 16500*

Alternate proof of sumdc 11981, without disjoint variable condition on N , x (longer because the statement is taylored to the proof sumdc 11981). (Contributed by BJ, 19-Feb-2022.)

|- ( ph -> M e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> A. x e. ( ZZ>= ` M )DECID x e. A )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> DECID N e. A )