P. 111 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

xpnnen 11001

The Cartesian product of the set of positive integers with itself is equinumerous to the set of positive integers. (Contributed by NM, 1-Aug-2004.)

|- ( NN X. NN ) ~~ NN

Theorem

xpomen 11002

The Cartesian product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.)

|- ( om X. om ) ~~ om

Theorem

xpct 11003

The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)

|- ( ( A ~<_ om /\ B ~<_ om ) -> ( A X. B ) ~<_ om )

Theorem

unennn 11004

The union of two disjoint countably infinite sets is countably infinite. (Contributed by Jim Kingdon, 13-May-2022.)

|- ( ( A ~~ NN /\ B ~~ NN /\ ( A i^i B ) = (/) ) -> ( A u. B ) ~~ NN )

Theorem

znnen 11005

The set of integers and the set of positive integers are equinumerous. Exercise 1 of [Gleason] p. 140. (Contributed by NM, 31-Jul-2004.)

|- ZZ ~~ NN

PART 5  GUIDES AND MISCELLANEA

5.1  Guides (conventions, explanations, and examples)

5.1.1  Conventions

This section describes the conventions we use. However, 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 11006

Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (aka "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 11060 (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 5593, 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). For this reason, we generally need to prove we are evaluating functions within their domains and avoid the reverse closure theorems of the Metamath Proof Explorer.
• 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.

Abbreviation	Mnenomic	Source	Expression	Syntax?	Example(s)
ap	apart	df-ap 7977		Yes	apadd1 8003, apne 8018

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

(Contributed by Jim Kingdon, 24-Feb-2020.)

|- ph   =>    |- ph

5.1.2  Definitional examples

Theorem

ex-or 11007

Example for ax-io 663. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

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

Theorem

ex-an 11008

Example for ax-ia1 104. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

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

Theorem

1kp2ke3k 11009

Example for df-dec 8787, 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 8787 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 11010

Example for df-fl 9580. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

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

Theorem

ex-ceil 11011

Example for df-ceil 9581. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-fac 11012

Example for df-fac 9983. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-bc 11013

Example for df-bc 10005. (Contributed by AV, 4-Sep-2021.)

|- ( 5 _C 3 ) = ; 1 0

Theorem

ex-dvds 11014

Example for df-dvds 10591: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

|- 3 || 6

Theorem

ex-gcd 11015

Example for df-gcd 10733. (Contributed by AV, 5-Sep-2021.)

|- ( -u 6 gcd 9 ) = 3

PART 6  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)

6.1  Mathboxes for user contributions

6.1.1  Mathbox guidelines

Theorem

mathbox 11016

(This theorem is a dummy placeholder for these guidelines. The name 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 set.mm.

For contributors:

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

Mathboxes are provided to help keep your work synchronized with changes in set.mm, but they shouldn't be depended on as a permanent archive. If you want to preserve your original contribution, it is your responsibility to keep your own copy of it along with the version of set.mm that works with it.

Guidelines:

1. If at all possible, please use only nullary class constants for new definitions.

2. Try to follow the style of the rest of set.mm. Each $p and $a statement must be immediately preceded with the comment that will be shown on its web page description. The metamath program command "write source set.mm /rewrap" will take care of wrapping comment lines and indentation conventions. All mathbox content will be on public display and should hopefully reflect the overall quality of the website.

3. Before submitting a revised mathbox, please make sure it verifies against the current set.mm.

4. Mathboxes should be independent i.e. the proofs should verify with all other mathboxes removed. If you need a theorem from another mathbox, that is fine (and encouraged), but let me know, so I can move the theorem to the main section. One way avoid undesired accidental use of other mathbox theorems is to develop your mathbox using a modified set.mm that has mathboxes removed.

Notes:

1. We may decide to move some theorems to the main part of set.mm for general use.

2. We may make changes to mathboxes to maintain the overall quality of set.mm. Normally we will let you know if a change might impact what you are working on.

3. If you use theorems from another user's mathbox, we don't provide assurance that they are based on correct or consistent $a statements. (If you find such a problem, please let us know so it can be corrected.) (Contributed by NM, 20-Feb-2007.) (New usage is discouraged.)

|- ph   =>    |- ph

6.2  Mathbox for BJ

6.2.1  Propositional calculus

Theorem

nnexmid 11017

Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but, of course, does not prove excluded middle) for any formula. (Contributed by BJ, 9-Oct-2019.)

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

Theorem

nndc 11018

Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but, of course, does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.)

|- -. -. DECID ph

Theorem

dcdc 11019

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 )

6.2.2  Predicate calculus

Theorem

bj-ex 11020*

Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1532 and 19.9ht 1575 or 19.23ht 1429). (Proof modification is discouraged.)

|- ( E. x ph -> ph )

Theorem

bj-hbalt 11021

Closed form of hbal 1409 (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 11022

Closed form of nfal 1511 (copied from set.mm). (Contributed by BJ, 2-May-2019.)

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

Theorem

spimd 11023

Deduction form of spim 1670. (Contributed by BJ, 17-Oct-2019.)

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

Theorem

2spim 11024*

Double substitution, as in spim 1670. (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 11025*

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 11026*

Version of ch2var 11025 with non-freeness hypotheses replaced by DV conditions. (Contributed by BJ, 17-Oct-2019.)

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

Theorem

bj-exlimmp 11027

Lemma for bj-vtoclgf 11033. (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 11028

Lemma for bj-vtoclgf 11033. (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 11029

A strengthening of sbiedh 1714 (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 11030

A strengthening of sbieh 1717 (same proof). (Contributed by BJ, 16-Dec-2019.)

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

Theorem

bj-sbime 11031

A strengthening of sbie 1718 (same proof). (Contributed by BJ, 16-Dec-2019.)

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

6.2.3  Extensionality

Various utility theorems using FOL and extensionality.

Theorem

bj-vtoclgft 11032

Weakening two hypotheses of vtoclgf 2670. (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 11033

Weakening two hypotheses of vtoclgf 2670. (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 11034

Lemma for elabgf 2748. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabgft1 11035

One implication of elabgf 2748, 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 11036

One implication of elabgf 2748. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabgf2 11037

One implication of elabgf 2748. (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 11038*

One implication of elabf 2749. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabf2 11039*

One implication of elabf 2749. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elab1 11040*

One implication of elab 2750. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elab2a 11041*

One implication of elab 2750. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabg2 11042*

One implication of elabg 2751. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

bj-rspgt 11043

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2711 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 11044

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2711 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 11045*

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 11046

Version of intss1 3680 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 11047

Version of intss1 3680 using a class abstraction and implicit substitution. Closed form of intmin3 3692. (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 11048

Version of elssuni 3658 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 11049

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 ) )

6.2.4  Dedidability 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 11051).

Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 11089).

Syntax

wdcin 11050

Syntax for decidability of a class in another.

wff A DECID_in B

Definition

df-dcin 11051*

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 11052

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 11053*

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 11054

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 11055

An upperset of integers is decidable in the integers. Reformulation of eluzdc 9006. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)

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

Theorem

sumdc2 11056*

Alternate proof of sumdc 10583, without DV condition on N , x (longer because the statement is taylored to the proof sumdc 10583). (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 )

6.2.5  Disjoint union

Theorem

djucllem 11057*

Lemma for djulcl 6664 and djurcl 6665. (Contributed by BJ, 4-Jul-2022.)

|- X e. _V   &    |- F = ( x e. _V |-> <. X , x >. )   =>    |- ( A e. B -> ( ( F |` B ) ` A ) e. ( { X } X. B ) )

Theorem

djulclALT 11058

Shortening of djulcl 6664 using djucllem 11057. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )

Theorem

djurclALT 11059

Shortening of djurcl 6665 using djucllem 11057. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )

6.2.6  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes

This section develops constructive Zermelo--Fraenkel set theory (CZF) on top of intuitionistic logic. It is a constructive theory in the sense that its logic is intuitionistic and it is predicative. "Predicative" means that new sets can be constructed only from already constructed sets. In particular, the axiom of separation ax-sep 3925 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 11132. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 3922 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 11234 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 11193. Similarly, the axiom of powerset ax-pow 3977 is not predicative (checking whether a set is included in another requires to universally quantifier over that "not yet constructed" set) and is replaced in CZF by the axiom of fullness or the axiom of subset collection ax-sscoll 11239.

In an intuitionistic context, the axiom of regularity is stated in IZF as well as in CZF as the axiom of set induction ax-setind 4319. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 11220.

For more details on CZF, a useful set of notes is

Peter Aczel and Michael Rathjen, CST Book draft. (available at http://www1.maths.leeds.ac.uk/~rathjen/book.pdf)

and an interesting article is

Michael Shulman, Comparing material and structural set theories, Annals of Pure and Applied Logic, Volume 170, Issue 4 (Apr. 2019), 465--504. (available at https://arxiv.org/abs/1808.05204)

I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results.

6.2.6.1  Bounded formulas

The present definition of bounded formulas emerged from a discussion on GitHub between Jim Kingdon, Mario Carneiro and I, started 23-Sept-2019 (see https://github.com/metamath/set.mm/issues/1173 and links therein).

In order to state certain axiom schemes of Constructive Zermelo–Fraenkel (CZF) set theory, like the axiom scheme of bounded (or restricted, or Δ₀) separation, it is necessary to distinguish certain formulas, called bounded (or restricted, or Δ₀) formulas. The necessity of considering bounded formulas also arises in several theories of bounded arithmetic, both classical or intuitonistic, for instance to state the axiom scheme of Δ₀-induction.

To formalize this in Metamath, there are several choices to make.

A first choice is to either create a new type for bounded formulas, or to create a predicate on formulas that indicates whether they are bounded. In the first case, one creates a new type "wff0" with a new set of metavariables (ph₀ ...) and an axiom "$a wff ph₀ " ensuring that bounded formulas are formulas, so that one can reuse existing theorems, and then axioms take the form "$a wff0 ( ph₀ -> ps₀ )", etc. In the second case, one introduces a predicate "BOUNDED " with the intended meaning that "BOUNDED ph " is a formula meaning that ph is a bounded formula. We choose the second option, since the first would complicate the grammar, risking to make it ambiguous. (TODO: elaborate.)

A second choice is to view "bounded" either as a syntactic or a semantic property. For instance, A. x T. is not syntactically bounded since it has an unbounded universal quantifier, but it is semantically bounded since it is equivalent to T. which is bounded. We choose the second option, so that formulas using defined symbols can be proved bounded.

A third choice is in the form of the axioms, either in closed form or in inference form. One cannot state all the axioms in closed form, especially ax-bd0 11061. Indeed, if we posited it in closed form, then we could prove for instance |- ( ph -> BOUNDED ph ) and |- ( -. ph -> BOUNDED ph ) which is problematic (with the law of excluded middle, this would entail that all formulas are bounded, but even without it, too many formulas could be proved bounded...). (TODO: elaborate.)

Having ax-bd0 11061 in inference form ensures that a formula can be proved bounded only if it is equivalent *for all values of the free variables* to a syntactically bounded one. The other axioms (ax-bdim 11062 through ax-bdsb 11070) can be written either in closed or inference form. The fact that ax-bd0 11061 is an inference is enough to ensure that the closed forms cannot be "exploited" to prove that some unbounded formulas are bounded. (TODO: check.) However, we state all the axioms in inference form to make it clear that we do not exploit any over-permissiveness.

Finally, note that our logic has no terms, only variables. Therefore, we cannot prove for instance that x e. om is a bounded formula. However, since om can be defined as "the y such that PHI" a proof using the fact that x e. om is bounded can be converted to a proof in iset.mm by replacing om with y everywhere and prepending the antecedent PHI, since x e. y is bounded by ax-bdel 11069. For a similar method, see bj-omtrans 11208.

Note that one cannot add an axiom |- BOUNDED x e. A since by bdph 11098 it would imply that every formula is bounded.

Syntax

wbd 11060

Syntax for the predicate BOUNDED.

wff BOUNDED ph

Axiom

ax-bd0 11061

If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.)

|- ( ph <-> ps )   =>    |- (BOUNDED ph -> BOUNDED ps )

Axiom

ax-bdim 11062

An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)

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

Axiom

ax-bdan 11063

The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)

|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph /\ ps )

Axiom

ax-bdor 11064

The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)

|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph \/ ps )

Axiom

ax-bdn 11065

The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)

|- BOUNDED ph   =>    |- BOUNDED -. ph

Axiom

ax-bdal 11066*

A bounded universal quantification of a bounded formula is bounded. Note the DV condition on x , y. (Contributed by BJ, 25-Sep-2019.)

|- BOUNDED ph   =>    |- BOUNDED A. x e. y ph

Axiom

ax-bdex 11067*

A bounded existential quantification of a bounded formula is bounded. Note the DV condition on x , y. (Contributed by BJ, 25-Sep-2019.)

|- BOUNDED ph   =>    |- BOUNDED E. x e. y ph

Axiom

ax-bdeq 11068

An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED x = y

Axiom

ax-bdel 11069

An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED x e. y

Axiom

ax-bdsb 11070

A formula resulting from proper substitution in a bounded formula is bounded. This probably cannot be proved from the other axioms, since neither the definiens in df-sb 1690, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED ph   =>    |- BOUNDED [ y / x ] ph

Theorem

bdeq 11071

Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)

|- ( ph <-> ps )   =>    |- (BOUNDED ph <-> BOUNDED ps )

Theorem

bd0 11072

A formula equivalent to a bounded one is bounded. See also bd0r 11073. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED ph   &    |- ( ph <-> ps )   =>    |- BOUNDED ps

Theorem

bd0r 11073

A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 11072) biconditional in the hypothesis, to work better with definitions ( ps is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED ph   &    |- ( ps <-> ph )   =>    |- BOUNDED ps

Theorem

bdbi 11074

A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph <-> ps )

Theorem

bdstab 11075

Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED ph   =>    |- BOUNDED STAB ph

Theorem

bddc 11076

Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED ph   =>    |- BOUNDED DECID ph

Theorem

bd3or 11077

A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

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

Theorem

bd3an 11078

A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

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

Theorem

bdth 11079

A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)

|- ph   =>    |- BOUNDED ph

Theorem

bdtru 11080

The truth value T. is bounded. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED T.

Theorem

bdfal 11081

The truth value F. is bounded. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED F.

Theorem

bdnth 11082

A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)

|- -. ph   =>    |- BOUNDED ph

Theorem

bdnthALT 11083

Alternate proof of bdnth 11082 not using bdfal 11081. Then, bdfal 11081 can be proved from this theorem, using fal 1294. The total number of proof steps would be 17 (for bdnthALT 11083) + 3 = 20, which is more than 8 (for bdfal 11081) + 9 (for bdnth 11082) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

|- -. ph   =>    |- BOUNDED ph

Theorem

bdxor 11084

The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph \/_ ps )

Theorem

bj-bdcel 11085*

Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)

|- BOUNDED y = A   =>    |- BOUNDED A e. x

Theorem

bdab 11086

Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED ph   =>    |- BOUNDED x e. { y | ph }

Theorem

bdcdeq 11087

Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)

|- BOUNDED ph   =>    |- BOUNDED CondEq ( x = y -> ph )

6.2.6.2  Bounded classes

In line with our definitions of classes as extensions of predicates, it is useful to define a predicate for bounded classes, which is done in df-bdc 11089. Note that this notion is only a technical device which can be used to shorten proofs of (semantic) boundedness of formulas.

As will be clear by the end of this subsection (see for instance bdop 11123), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, |- BOUNDED ph => |- BOUNDED <. { x | ph } , ( { y , suc z } X. <. t , (/) >. ) >.. The proofs are long since one has to prove boundedness at each step of the construction, without being able to prove general theorems like |- BOUNDED A => |- BOUNDED { A }.

Syntax

wbdc 11088

Syntax for the predicate BOUNDED.

wff BOUNDED A

Definition

df-bdc 11089*

Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)

|- (BOUNDED A <-> A. xBOUNDED x e. A )

Theorem

bdceq 11090

Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)

|- A = B   =>    |- (BOUNDED A <-> BOUNDED B )

Theorem

bdceqi 11091

A class equal to a bounded one is bounded. Note the use of ax-ext 2067. See also bdceqir 11092. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED A   &    |- A = B   =>    |- BOUNDED B

Theorem

bdceqir 11092

A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 11091) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 11073). (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED A   &    |- B = A   =>    |- BOUNDED B

Theorem

bdel 11093*

The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)

|- (BOUNDED A -> BOUNDED x e. A )

Theorem

bdeli 11094*

Inference associated with bdel 11093. Its converse is bdelir 11095. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED A   =>    |- BOUNDED x e. A

Theorem

bdelir 11095*

Inference associated with df-bdc 11089. Its converse is bdeli 11094. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED x e. A   =>    |- BOUNDED A

Theorem

bdcv 11096

A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED x

Theorem

bdcab 11097

A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)

|- BOUNDED ph   =>    |- BOUNDED { x | ph }

Theorem

bdph 11098

A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)

|- BOUNDED { x | ph }   =>    |- BOUNDED ph

Theorem

bds 11099*

Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 11070; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 11070. (Contributed by BJ, 19-Nov-2019.)

|- BOUNDED ph   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- BOUNDED ps

Theorem

bdcrab 11100*

A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)

|- BOUNDED A   &    |- BOUNDED ph   =>    |- BOUNDED { x e. A | ph }