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Theorem List for Intuitionistic Logic Explorer - 16901-17000 *Has distinct variable group(s)

Type

Label

Description

Statement

Theorem

konigsberglem2 16901

Lemma 2 for konigsberg 16905: 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 16902

Lemma 3 for konigsberg 16905: 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 16903*

Lemma 4 for konigsberg 16905: 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} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}

Theorem

konigsberglem5 16904*

Lemma 5 for konigsberg 16905: 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 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})

Theorem

konigsberg 16905

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 16893 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 16906

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 17009 (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 6084, 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 5727 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-ap 8913	#	Yes	apadd1 8939, apne 8954
g	with "is a set" condition			No	1stvalg 6376, brtposg 6525, setsmsbasg 15671
m	inhabited (from "member")		∃𝑥𝑥 ∈ 𝐴	No	r19.2m 3614, negm 10025, ctm 7450, basmex 13464
pap	apartness predicate	df-pap 7609	Ap	Yes	papeq1 7610, papirr 7612
seq3, sum3	recursive sequence	df-seqfrec 10900		Yes	seq3-1 10914, fsum3 12173
tap	tight apartness	df-tap 7616	TAp	Yes	df-tap 7616

Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7616.

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

𝜑 ⇒ 𝜑

13.1.2 Definitional examples

Theorem

ex-or 16907

Example for ax-io 721. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

(2 = 3 ∨ 4 = 4)

Theorem

ex-an 16908

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

(2 = 2 ∧ 3 = 3)

Theorem

1kp2ke3k 16909

Example for df-dec 9783, 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 9783 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.)

(1000 + 2000) = 3000

Theorem

ex-fl 16910

Example for df-fl 10716. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2)

Theorem

ex-ceil 16911

Example for df-ceil 10717. (Contributed by AV, 4-Sep-2021.)

((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)

Theorem

ex-exp 16912

Example for df-exp 10991. (Contributed by AV, 4-Sep-2021.)

((5↑2) = 25 ∧ (-3↑-2) = (1 / 9))

Theorem

ex-fac 16913

Example for df-fac 11180. (Contributed by AV, 4-Sep-2021.)

(!‘5) = 120

Theorem

ex-bc 16914

Example for df-bc 11202. (Contributed by AV, 4-Sep-2021.)

(5C3) = 10

Theorem

ex-dvds 16915

Example for df-dvds 12574: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

3 ∥ 6

Theorem

ex-gcd 16916

Example for df-gcd 12750. (Contributed by AV, 5-Sep-2021.)

(-6 gcd 9) = 3

PART 14 SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

14.1 Mathboxes for user contributions

14.1.1 Mathbox guidelines

Theorem

mathbox 16917

(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.)

𝜑 ⇒ 𝜑

14.2 Mathbox for Matthew House

Theorem

depindlem1 16918*

Lemma for depind 16921. (Contributed by Matthew House, 14-Apr-2026.)

(𝜑 → 𝑃:ℕ₀⟶V) & (𝜑 → 𝐴 ∈ (𝑃‘0)) & (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1))) & 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))) ⇒ (𝜑 → (𝐹:ℕ₀⟶V ∧ (𝐹‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ₀ (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛))))

Theorem

depindlem2 16919*

Lemma for depind 16921. (Contributed by Matthew House, 14-Apr-2026.)

(𝜑 → 𝑃:ℕ₀⟶V) & (𝜑 → 𝐴 ∈ (𝑃‘0)) & (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1))) & 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))) ⇒ (𝜑 → 𝐹 ∈ X𝑛 ∈ ℕ₀ (𝑃‘𝑛))

Theorem

depindlem3 16920*

Lemma for depind 16921. (Contributed by Matthew House, 14-Apr-2026.)

(𝜑 → 𝑃:ℕ₀⟶V) & (𝜑 → 𝐴 ∈ (𝑃‘0)) & (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1))) & 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))) ⇒ (𝜑 → ∀𝑓 ∈ X 𝑛 ∈ ℕ₀ (𝑃‘𝑛)(((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ₀ (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))) → 𝑓 = 𝐹))

Theorem

depind 16921*

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

(𝜑 → 𝑃:ℕ₀⟶V) & (𝜑 → 𝐴 ∈ (𝑃‘0)) & (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1))) ⇒ (𝜑 → ∃!𝑓 ∈ X 𝑛 ∈ ℕ₀ (𝑃‘𝑛)((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ₀ (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))))

Theorem

lealltlt1 16922*

Alternative definition for ≤ on real numbers. (Contributed by Matthew House, 29-Jun-2026.)

((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ (𝑥 < 𝐴 → 𝑥 < 𝐵)))

Theorem

lealltlt2 16923*

Alternative definition for ≤ on real numbers. (Contributed by Matthew House, 29-Jun-2026.)

((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ (𝐵 < 𝑥 → 𝐴 < 𝑥)))

Theorem

dichmul0orlem1 16924