P. 167 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

eulerpathprum 16601*

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 ∈ UMGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃 ∧ 𝑉 ∈ Fin) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})

Theorem

eulerpathum 16602*

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 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {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 16608. konigsberg 16614 shows that the Königsberg graph has no Eulerian path, thus the Königsberg Bridge problem has no solution.

Theorem

konigsbergvtx 16603

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 16604

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 16605*

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 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o}

Theorem

konigsbergssiedgwpren 16606*

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 ∈ Word V ∧ 𝐵 ∈ Word V ∧ 𝐸 = (𝐴 ++ 𝐵)) → 𝐴 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ 𝑥 ≈ 2_o})

Theorem

konigsbergssiedgwen 16607*

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 ∈ Word V ∧ 𝐵 ∈ Word V ∧ 𝐸 = (𝐴 ++ 𝐵)) → 𝐴 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1_o ∨ 𝑥 ≈ 2_o)})

Theorem

konigsbergumgr 16608

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 ∈ UMGraph

Theorem

konigsberglem1 16609

Lemma 1 for konigsberg 16614: 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 16610

Lemma 2 for konigsberg 16614: 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 16611

Lemma 3 for konigsberg 16614: 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 16612*

Lemma 4 for konigsberg 16614: 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 16613*

Lemma 5 for konigsberg 16614: 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 16614

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 16602 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 16615

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 16708 (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 6057, 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 5707 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 8873	#	Yes	apadd1 8899, apne 8914
g	with "is a set" condition			No	1stvalg 6349, brtposg 6498, setsmsbasg 15470
m	inhabited (from "member")		∃𝑥𝑥 ∈ 𝐴	No	r19.2m 3600, negm 9965, ctm 7413, basmex 13356
pap	apartness predicate	df-pap 7572	Ap	Yes	papeq1 7573, papirr 7575
seq3, sum3	recursive sequence	df-seqfrec 10834		Yes	seq3-1 10848, fsum3 12098
tap	tight apartness	df-tap 7579	TAp	Yes	df-tap 7579

• Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7579.

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

⊢ 𝜑    ⇒   ⊢ 𝜑

13.1.2  Definitional examples

Theorem

ex-or 16616

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

⊢ (2 = 3 ∨ 4 = 4)

Theorem

ex-an 16617

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

⊢ (2 = 2 ∧ 3 = 3)

Theorem

1kp2ke3k 16618

Example for df-dec 9728, 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 9728 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 16619

Example for df-fl 10654. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2)

Theorem

ex-ceil 16620

Example for df-ceil 10655. (Contributed by AV, 4-Sep-2021.)

⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)

Theorem

ex-exp 16621

Example for df-exp 10925. (Contributed by AV, 4-Sep-2021.)

⊢ ((5↑2) = ;25 ∧ (-3↑-2) = (1 / 9))

Theorem

ex-fac 16622

Example for df-fac 11113. (Contributed by AV, 4-Sep-2021.)

⊢ (!‘5) = ;;120

Theorem

ex-bc 16623

Example for df-bc 11135. (Contributed by AV, 4-Sep-2021.)

⊢ (5C3) = ;10

Theorem

ex-dvds 16624

Example for df-dvds 12499: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

⊢ 3 ∥ 6

Theorem

ex-gcd 16625

Example for df-gcd 12675. (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 16626

(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 16627*

Lemma for depind 16630. (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 16628*

Lemma for depind 16630. (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 16629*

Lemma for depind 16630. (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 16630*

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))))

14.3  Mathbox for BJ

14.3.1  Propositional calculus

Theorem

bj-nnsn 16631

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

⊢ ((𝜑 → ¬ 𝜓) ↔ (¬ ¬ 𝜑 → ¬ 𝜓))

Theorem

bj-nnor 16632

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

⊢ (¬ ¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓))

Theorem

bj-nnim 16633

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

⊢ (¬ ¬ (𝜑 → 𝜓) → (𝜑 → ¬ ¬ 𝜓))

Theorem

bj-nnan 16634

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

⊢ (¬ ¬ (𝜑 ∧ 𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))

Theorem

bj-nnclavius 16635

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

⊢ ((¬ 𝜑 → 𝜑) → ¬ ¬ 𝜑)

Theorem

bj-imnimnn 16636

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 16635 as its last step. (Contributed by BJ, 27-Oct-2024.)

⊢ (𝜑 → 𝜓)    &   ⊢ (¬ 𝜑 → 𝜓)    ⇒   ⊢ ¬ ¬ 𝜓

14.3.1.1  Stable formulas

Some of the following theorems, like bj-sttru 16638 or bj-stfal 16640 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 16637

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

⊢ (𝜑 → STAB 𝜑)

Theorem

bj-sttru 16638

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

⊢ STAB ⊤

Theorem

bj-fast 16639

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

⊢ (¬ 𝜑 → STAB 𝜑)

Theorem

bj-stfal 16640

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

⊢ STAB ⊥

Theorem

bj-nnst 16641

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 16886 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 𝜑

Theorem

bj-nnbist 16642

If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if 𝜑 is a classical tautology, then ¬ ¬ 𝜑 is an intuitionistic tautology. Therefore, if 𝜑 is a classical tautology, then 𝜑 is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 16655). (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ ¬ 𝜑 → (STAB 𝜑 ↔ 𝜑))

Theorem

bj-stst 16643

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 𝜑 ↔ STAB 𝜑)

Theorem

bj-stim 16644

A conjunction with a stable consequent is stable. See stabnot 841 for negation , bj-stan 16645 for conjunction , and bj-stal 16647 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

⊢ (STAB 𝜓 → STAB (𝜑 → 𝜓))

Theorem

bj-stan 16645

The conjunction of two stable formulas is stable. See bj-stim 16644 for implication, stabnot 841 for negation, and bj-stal 16647 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

⊢ ((STAB 𝜑 ∧ STAB 𝜓) → STAB (𝜑 ∧ 𝜓))

Theorem

bj-stand 16646

The conjunction of two stable formulas is stable. Deduction form of bj-stan 16645. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 16645 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

⊢ (𝜑 → STAB 𝜓)    &   ⊢ (𝜑 → STAB 𝜒)    ⇒   ⊢ (𝜑 → STAB (𝜓 ∧ 𝜒))

Theorem

bj-stal 16647

The universal quantification of a stable formula is stable. See bj-stim 16644 for implication, stabnot 841 for negation, and bj-stan 16645 for conjunction. (Contributed by BJ, 24-Nov-2023.)

⊢ (∀𝑥STAB 𝜑 → STAB ∀𝑥𝜑)

Theorem

bj-pm2.18st 16648

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

⊢ (STAB 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))

Theorem

bj-con1st 16649

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

⊢ (STAB 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))

14.3.1.2  Decidable formulas

Theorem

bj-trdc 16650

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

⊢ (𝜑 → DECID 𝜑)

Theorem

bj-dctru 16651

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

⊢ DECID ⊤

Theorem

bj-fadc 16652

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

⊢ (¬ 𝜑 → DECID 𝜑)

Theorem

bj-dcfal 16653

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

⊢ DECID ⊥

Theorem

bj-dcstab 16654

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

⊢ (DECID 𝜑 → STAB 𝜑)

Theorem

bj-nnbidc 16655

If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 16642. (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ ¬ 𝜑 → (DECID 𝜑 ↔ 𝜑))

Theorem

bj-nndcALT 16656

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

⊢ ¬ ¬ DECID 𝜑

Theorem

bj-dcdc 16657

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 𝜑 ↔ DECID 𝜑)

Theorem

bj-stdc 16658

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 𝜑 ↔ DECID 𝜑)

Theorem

bj-dcst 16659

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

⊢ (DECID STAB 𝜑 ↔ STAB 𝜑)

14.3.2  Predicate calculus

Theorem

bj-ex 16660*

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.)

⊢ (∃𝑥𝜑 → 𝜑)

Theorem

bj-hbalt 16661

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

⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Theorem

bj-nfalt 16662

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

⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)

Theorem

spimd 16663

Deduction form of spim 1787. (Contributed by BJ, 17-Oct-2019.)

⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Theorem

2spim 16664*

Double substitution, as in spim 1787. (Contributed by BJ, 17-Oct-2019.)

⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑧𝜒    &   ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑧∀𝑥𝜓 → 𝜒)

Theorem

ch2var 16665*

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

⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓

Theorem

ch2varv 16666*

Version of ch2var 16665 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)

⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓

Theorem

bj-exlimmp 16667

Lemma for bj-vtoclgf 16674. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓))

Theorem

bj-exlimmpi 16668

Lemma for bj-vtoclgf 16674. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    &   ⊢ (𝜒 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)

Theorem

bj-sbimedh 16669

A strengthening of sbiedh 1836 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒))

Theorem

bj-sbimeh 16670

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

⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

Theorem

bj-sbime 16671

A strengthening of sbie 1840 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

14.3.3  Set theorey miscellaneous

Theorem

bj-el2oss1o 16672

Shorter proof of el2oss1o 6689 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 2_o → 𝐴 ⊆ 1_o)

14.3.4  Extensionality

Various utility theorems using FOL and extensionality.

Theorem

bj-vtoclgft 16673

Weakening two hypotheses of vtoclgf 2875. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓))

Theorem

bj-vtoclgf 16674

Weakening two hypotheses of vtoclgf 2875. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Theorem

elabgf0 16675

Lemma for elabgf 2962. (Contributed by BJ, 21-Nov-2019.)

⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))

Theorem

elabgft1 16676

One implication of elabgf 2962, in closed form. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))

Theorem

elabgf1 16677

One implication of elabgf 2962. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Theorem

elabgf2 16678

One implication of elabgf 2962. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))

Theorem

elabf1 16679*

One implication of elabf 2963. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Theorem

elabf2 16680*

One implication of elabf 2963. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Theorem

elab1 16681*

One implication of elab 2964. (Contributed by BJ, 21-Nov-2019.)

⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Theorem

elab2a 16682*

One implication of elab 2964. (Contributed by BJ, 21-Nov-2019.)

⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Theorem

elabg2 16683*

One implication of elabg 2966. (Contributed by BJ, 21-Nov-2019.)

⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))

Theorem

bj-rspgt 16684

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2920 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)))

Theorem

bj-rspg 16685

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2920 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))

Theorem

cbvrald 16686*

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

⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))

Theorem

bj-intabssel 16687

Version of intss1 3969 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))

Theorem

bj-intabssel1 16688

Version of intss1 3969 using a class abstraction and implicit substitution. Closed form of intmin3 3981. (Contributed by BJ, 29-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))

Theorem

bj-elssuniab 16689

Version of elssuni 3947 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)

⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → 𝐴 ⊆ ∪ {𝑥 ∣ 𝜑}))

Theorem

bj-sseq 16690

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.)

⊢ (𝜑 → (𝜓 ↔ 𝐴 ⊆ 𝐵))    &   ⊢ (𝜑 → (𝜒 ↔ 𝐵 ⊆ 𝐴))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 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 ∀𝑥 ∈ 𝐵DECID 𝑥 ∈ 𝐴 (see df-dcin 16692).

Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 16737).

Syntax

wdcin 16691

Syntax for decidability of a class in another.

wff 𝐴 DECID_in 𝐵

Definition

df-dcin 16692*

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

⊢ (𝐴 DECID_in 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)

Theorem

decidi 16693

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

⊢ (𝐴 DECID_in 𝐵 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))

Theorem

decidr 16694*

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

⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))    ⇒   ⊢ (𝜑 → 𝐴 DECID_in 𝐵)

Theorem

decidin 16695

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.)

⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐴 DECID_in 𝐵)    &   ⊢ (𝜑 → 𝐵 DECID_in 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 DECID_in 𝐶)

Theorem

uzdcinzz 16696

An upperset of integers is decidable in the integers. Reformulation of eluzdc 9960. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)

⊢ (𝑀 ∈ ℤ → (ℤ_≥‘𝑀) DECID_in ℤ)

Theorem

sumdc2 16697*

Alternate proof of sumdc 12068, without disjoint variable condition on 𝑁, 𝑥 (longer because the statement is taylored to the proof sumdc 12068). (Contributed by BJ, 19-Feb-2022.)

⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))    &   ⊢ (𝜑 → ∀𝑥 ∈ (ℤ_≥‘𝑀)DECID 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴)

14.3.6  Disjoint union

Theorem

djucllem 16698*

Lemma for djulcl 7355 and djurcl 7356. (Contributed by BJ, 4-Jul-2022.)

⊢ 𝑋 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)    ⇒   ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))

Theorem

djulclALT 16699

Shortening of djulcl 7355 using djucllem 16698. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵))

Theorem

djurclALT 16700

Shortening of djurcl 7356 using djucllem 16698. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵))