P. 162 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

reltrls 16101

The set (Trails ` G ) of all trails on G is a set of pairs by our definition of a trail, and so is a relation. (Contributed by AV, 29-Oct-2021.)

|- Rel (Trails ` G )

Theorem

trlsfvalg 16102*

The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.)

|- ( G e. V -> (Trails ` G ) = { <. f , p >. | ( f (Walks ` G ) p /\ Fun `' f ) } )

Theorem

trlsv 16103

The classes involved in a trail are sets. (Contributed by Jim Kingdon, 7-Feb-2026.)

|- ( F (Trails ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )

Theorem

istrl 16104

Conditions for a pair of classes/functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.)

|- ( F (Trails ` G ) P <-> ( F (Walks ` G ) P /\ Fun `' F ) )

Theorem

trliswlk 16105

A trail is a walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 29-Oct-2021.)

|- ( F (Trails ` G ) P -> F (Walks ` G ) P )

Theorem

trlf1 16106

The enumeration F of a trail <. F , P >. is injective. (Contributed by AV, 20-Feb-2021.) (Proof shortened by AV, 29-Oct-2021.)

|- I = (iEdg ` G )   =>    |- ( F (Trails ` G ) P -> F : ( 0..^ (♯ ` F ) ) -1-1-> dom I )

Theorem

trlreslem 16107

Lemma for trlres 16108. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- H = ( F prefix N )   =>    |- ( ph -> H : ( 0..^ (♯ ` H ) ) -1-1-onto-> dom ( I |` ( F " ( 0..^ N ) ) ) )

Theorem

trlres 16108

The restriction <. H , Q >. of a trail <. F , P >. to an initial segment of the trail (of length N) forms a trail on the subgraph S consisting of the edges in the initial segment. (Contributed by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> F (Trails ` G ) P )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- H = ( F prefix N )   &    |- ( ph -> (Vtx ` S ) = V )   &    |- ( ph -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- Q = ( P |` ( 0 ... N ) )   =>    |- ( ph -> H (Trails ` S ) Q )

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 16109

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 16199 (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 6006, 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 5661 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 7442, df-ap 8737		Yes	apadd1 8763, apne 8778
g	with "is a set" condition			No	1stvalg 6294, brtposg 6406, setsmsbasg 15161
m	inhabited (from "member")		E. x x e. A	No	r19.2m 3578, negm 9818, ctm 7284, basmex 13100
seq3, sum3	recursive sequence	df-seqfrec 10678		Yes	seq3-1 10692, fsum3 11906
tap	tight apartness	df-tap 7444		Yes	df-tap 7444

• Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7444.

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

|- ph   =>    |- ph

13.1.2  Definitional examples

Theorem

ex-or 16110

Example for ax-io 714. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

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

Theorem

ex-an 16111

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

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

Theorem

1kp2ke3k 16112

Example for df-dec 9587, 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 9587 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 16113

Example for df-fl 10498. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

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

Theorem

ex-ceil 16114

Example for df-ceil 10499. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-exp 16115

Example for df-exp 10769. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-fac 16116

Example for df-fac 10956. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-bc 16117

Example for df-bc 10978. (Contributed by AV, 4-Sep-2021.)

|- ( 5 _C 3 ) = ; 1 0

Theorem

ex-dvds 16118

Example for df-dvds 12307: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

|- 3 || 6

Theorem

ex-gcd 16119

Example for df-gcd 12483. (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 16120

(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 BJ

14.2.1  Propositional calculus

Theorem

bj-nnsn 16121

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 16122

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

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

Theorem

bj-nnim 16123

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 16124

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 16125

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

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

Theorem

bj-imnimnn 16126

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 16125 as its last step. (Contributed by BJ, 27-Oct-2024.)

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

14.2.1.1  Stable formulas

Some of the following theorems, like bj-sttru 16128 or bj-stfal 16130 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 16127

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

|- ( ph -> STAB ph )

Theorem

bj-sttru 16128

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

|- STAB T.

Theorem

bj-fast 16129

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

|- ( -. ph -> STAB ph )

Theorem

bj-stfal 16130

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

|- STAB F.

Theorem

bj-nnst 16131

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 16377 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 16132

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 16145). (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stst 16133

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 16134

A conjunction with a stable consequent is stable. See stabnot 838 for negation , bj-stan 16135 for conjunction , and bj-stal 16137 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stan 16135

The conjunction of two stable formulas is stable. See bj-stim 16134 for implication, stabnot 838 for negation, and bj-stal 16137 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stand 16136

The conjunction of two stable formulas is stable. Deduction form of bj-stan 16135. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 16135 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 16137

The universal quantification of a stable formula is stable. See bj-stim 16134 for implication, stabnot 838 for negation, and bj-stan 16135 for conjunction. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-pm2.18st 16138

Clavius law for stable formulas. See pm2.18dc 860. (Contributed by BJ, 4-Dec-2023.)

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

Theorem

bj-con1st 16139

Contraposition when the antecedent is a negated stable proposition. See con1dc 861. (Contributed by BJ, 11-Nov-2024.)

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

14.2.1.2  Decidable formulas

Theorem

bj-trdc 16140

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

|- ( ph -> DECID ph )

Theorem

bj-dctru 16141

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

|- DECID T.

Theorem

bj-fadc 16142

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

|- ( -. ph -> DECID ph )

Theorem

bj-dcfal 16143

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

|- DECID F.

Theorem

bj-dcstab 16144

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

|- (DECID ph -> STAB ph )

Theorem

bj-nnbidc 16145

If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 16132. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-nndcALT 16146

Alternate proof of nndc 856. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)

|- -. -. DECID ph

Theorem

bj-dcdc 16147

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 16148

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 16149

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 )

14.2.2  Predicate calculus

Theorem

bj-ex 16150*

Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1644 and 19.9ht 1687 or 19.23ht 1543). (Proof modification is discouraged.)

|- ( E. x ph -> ph )

Theorem

bj-hbalt 16151

Closed form of hbal 1523 (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 16152

Closed form of nfal 1622 (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 16153

Deduction form of spim 1784. (Contributed by BJ, 17-Oct-2019.)

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

Theorem

2spim 16154*

Double substitution, as in spim 1784. (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 16155*

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 16156*

Version of ch2var 16155 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 16157

Lemma for bj-vtoclgf 16164. (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 16158

Lemma for bj-vtoclgf 16164. (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 16159

A strengthening of sbiedh 1833 (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 16160

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

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

Theorem

bj-sbime 16161

A strengthening of sbie 1837 (same proof). (Contributed by BJ, 16-Dec-2019.)

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

14.2.3  Set theorey miscellaneous

Theorem

bj-el2oss1o 16162

Shorter proof of el2oss1o 6597 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

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

14.2.4  Extensionality

Various utility theorems using FOL and extensionality.

Theorem

bj-vtoclgft 16163

Weakening two hypotheses of vtoclgf 2859. (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 16164

Weakening two hypotheses of vtoclgf 2859. (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 16165

Lemma for elabgf 2945. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabgft1 16166

One implication of elabgf 2945, 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 16167

One implication of elabgf 2945. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabgf2 16168

One implication of elabgf 2945. (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 16169*

One implication of elabf 2946. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabf2 16170*

One implication of elabf 2946. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elab1 16171*

One implication of elab 2947. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elab2a 16172*

One implication of elab 2947. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabg2 16173*

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

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

Theorem

bj-rspgt 16174

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2904 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 16175

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2904 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 16176*

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 16177

Version of intss1 3938 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 16178

Version of intss1 3938 using a class abstraction and implicit substitution. Closed form of intmin3 3950. (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 16179

Version of elssuni 3916 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 16180

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 ) )

14.2.5  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 16182).

Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 16228).

Syntax

wdcin 16181

Syntax for decidability of a class in another.

wff A DECID_in B

Definition

df-dcin 16182*

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 16183

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 16184*

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 16185

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 16186

An upperset of integers is decidable in the integers. Reformulation of eluzdc 9813. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)

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

Theorem

sumdc2 16187*

Alternate proof of sumdc 11877, without disjoint variable condition on N , x (longer because the statement is taylored to the proof sumdc 11877). (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 )

14.2.6  Disjoint union

Theorem

djucllem 16188*

Lemma for djulcl 7226 and djurcl 7227. (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 16189

Shortening of djulcl 7226 using djucllem 16188. (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 16190

Shortening of djurcl 7227 using djucllem 16188. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

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

14.2.7  Miscellaneous

Theorem

funmptd 16191

The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5355, then prove funmptd 16191 from it, and then prove funmpt 5356 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)

|- ( ph -> F = ( x e. A |-> B ) )   =>    |- ( ph -> Fun F )

Theorem

fnmptd 16192*

The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)

|- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ( ph /\ x e. A ) -> B e. V )   =>    |- ( ph -> F Fn A )

Theorem

if0ab 16193*

Expression of a conditional class as a class abstraction when the False alternative is the empty class: in that case, the conditional class is the extension, in the True alternative, of the condition.

Remark: a consequence which could be formalized is the inclusion |- if ( ph , A , (/) ) C_ A and therefore, using elpwg 3657, |- ( A e. V -> if ( ph , A , (/) ) e. ~P A ), from which fmelpw1o 7440 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)

|- if ( ph , A , (/) ) = { x e. A | ph }

Theorem

bj-charfun 16194*

Properties of the characteristic function on the class X of the class A. (Contributed by BJ, 15-Aug-2024.)

|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   =>    |- ( ph -> ( ( F : X --> ~P 1o /\ ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o ) /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )

Theorem

bj-charfundc 16195*

Properties of the characteristic function on the class X of the class A, provided membership in A is decidable in X. (Contributed by BJ, 6-Aug-2024.)

|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   &    |- ( ph -> A. x e. X DECID x e. A )   =>    |- ( ph -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )

Theorem

bj-charfundcALT 16196*

Alternate proof of bj-charfundc 16195. It was expected to be much shorter since it uses bj-charfun 16194 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   &    |- ( ph -> A. x e. X DECID x e. A )   =>    |- ( ph -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )

Theorem

bj-charfunr 16197*

If a class A has a "weak" characteristic function on a class X, then negated membership in A is decidable (in other words, membership in A is testable) in X.

The hypothesis imposes that X be a set. As usual, it could be formulated as |- ( ph -> ( F : X --> om /\ ... ) ) to deal with general classes, but that extra generality would not make the theorem much more useful.

The theorem would still hold if the codomain of f were any class with testable equality to the point where ( X \ A ) is sent. (Contributed by BJ, 6-Aug-2024.)

|- ( ph -> E. f e. ( om ^m X ) ( A. x e. ( X i^i A ) ( f ` x ) =/= (/) /\ A. x e. ( X \ A ) ( f ` x ) = (/) ) )   =>    |- ( ph -> A. x e. X DECID -. x e. A )

Theorem

bj-charfunbi 16198*

In an ambient set X, if membership in A is stable, then it is decidable if and only if A has a characteristic function.

This characterization can be applied to singletons when the set X has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.)

|- ( ph -> X e. V )   &    |- ( ph -> A. x e. X STAB x e. A )   =>    |- ( ph -> ( A. x e. X DECID x e. A <-> E. f e. ( 2o ^m X ) ( A. x e. ( X i^i A ) ( f ` x ) = 1o /\ A. x e. ( X \ A ) ( f ` x ) = (/) ) ) )

14.2.8  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 4202 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 16271. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4199 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 16369 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 16328. Similarly, the axiom of powerset ax-pow 4258 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 16374.

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 4629. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 16355.

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 16355)

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. https://doi.org/10.48550/arXiv.1808.05204 16355

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

14.2.8.1  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 intuitionistic, 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 16200. 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 16200 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 16201 through ax-bdsb 16209) can be written either in closed or inference form. The fact that ax-bd0 16200 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 16208. For a similar method, see bj-omtrans 16343.

Note that one cannot add an axiom |- BOUNDED x e. A since by bdph 16237 it would imply that every formula is bounded.

Syntax

wbd 16199

Syntax for the predicate BOUNDED.

wff BOUNDED ph

Axiom

ax-bd0 16200

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 )