P. 159 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

upgr1eopdc 15801

A pseudograph with one edge. Such a graph is actually a simple pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)

|- ( ph -> V e. W )   &    |- ( ph -> A e. X )   &    |- ( ph -> B e. V )   &    |- ( ph -> C e. V )   &    |- ( ph -> DECID B = C )   =>    |- ( ph -> <. V , { <. A , { B , C } >. } >. e. UPGraph )

Theorem

upgrun 15802

The union U of two pseudographs G and H with the same vertex set V is a pseudograph with the vertex V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)

|- ( ph -> G e. UPGraph )   &    |- ( ph -> H e. UPGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   &    |- ( ph -> U e. W )   &    |- ( ph -> (Vtx ` U ) = V )   &    |- ( ph -> (iEdg ` U ) = ( E u. F ) )   =>    |- ( ph -> U e. UPGraph )

Theorem

upgrunop 15803

The union of two pseudographs (with the same vertex set): If <. V , E >. and <. V , F >. are pseudographs, then <. V , E u. F >. is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)

|- ( ph -> G e. UPGraph )   &    |- ( ph -> H e. UPGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   =>    |- ( ph -> <. V , ( E u. F ) >. e. UPGraph )

Theorem

umgrun 15804

The union U of two multigraphs G and H with the same vertex set V is a multigraph with the vertex V and the union ( E u. F ) of the (indexed) edges. (Contributed by AV, 25-Nov-2020.)

|- ( ph -> G e. UMGraph )   &    |- ( ph -> H e. UMGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   &    |- ( ph -> U e. W )   &    |- ( ph -> (Vtx ` U ) = V )   &    |- ( ph -> (iEdg ` U ) = ( E u. F ) )   =>    |- ( ph -> U e. UMGraph )

Theorem

umgrunop 15805

The union of two multigraphs (with the same vertex set): If <. V , E >. and <. V , F >. are multigraphs, then <. V , E u. F >. is a multigraph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)

|- ( ph -> G e. UMGraph )   &    |- ( ph -> H e. UMGraph )   &    |- E = (iEdg ` G )   &    |- F = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> ( dom E i^i dom F ) = (/) )   =>    |- ( ph -> <. V , ( E u. F ) >. e. UMGraph )

12.2.3  Loop-free graphs

For a hypergraph, the property to be "loop-free" is expressed by I : dom I --> E with E = { x e. ~P V | 2o ~<_ x } and I = (iEdg ` G ). E is the set of edges which connect at least two vertices.

Theorem

umgrislfupgrenlem 15806

Lemma for umgrislfupgrdom 15807. (Contributed by AV, 27-Jan-2021.)

|- ( { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } i^i { x e. ~P V | 2o ~<_ x } ) = { x e. ~P V | x ~~ 2o }

Theorem

umgrislfupgrdom 15807*

A multigraph is a loop-free pseudograph. (Contributed by AV, 27-Jan-2021.)

|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( G e. UMGraph <-> ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) )

Theorem

lfgredg2dom 15808*

An edge of a loop-free graph has at least two ends. (Contributed by AV, 23-Feb-2021.)

|- I = (iEdg ` G )   &    |- A = dom I   &    |- E = { x e. ~P V | 2o ~<_ x }   =>    |- ( ( I : A --> E /\ X e. A ) -> 2o ~<_ ( I ` X ) )

Theorem

lfgrnloopen 15809*

A loop-free graph has no loops. (Contributed by AV, 23-Feb-2021.)

|- I = (iEdg ` G )   &    |- A = dom I   &    |- E = { x e. ~P V | 2o ~<_ x }   =>    |- ( I : A --> E -> { x e. A | ( I ` x ) ~~ 1o } = (/) )

12.2.4  Edges as subsets of vertices of graphs

Theorem

uhgredgiedgb 15810*

In a hypergraph, a set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.)

|- I = (iEdg ` G )   =>    |- ( G e. UHGraph -> ( E e. (Edg ` G ) <-> E. x e. dom I E = ( I ` x ) ) )

Theorem

uhgriedg0edg0 15811

A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 8-Dec-2021.)

|- ( G e. UHGraph -> ( (Edg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )

Theorem

uhgredgm 15812*

An edge of a hypergraph is an inhabited subset of vertices. (Contributed by AV, 28-Nov-2020.)

|- ( ( G e. UHGraph /\ E e. (Edg ` G ) ) -> ( E e. ~P (Vtx ` G ) /\ E. x x e. E ) )

Theorem

edguhgr 15813

An edge of a hypergraph is a subset of vertices. (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 28-Nov-2020.)

|- ( ( G e. UHGraph /\ E e. (Edg ` G ) ) -> E e. ~P (Vtx ` G ) )

Theorem

uhgredgrnv 15814

An edge of a hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 4-Jun-2021.)

|- ( ( G e. UHGraph /\ E e. (Edg ` G ) /\ N e. E ) -> N e. (Vtx ` G ) )

Theorem

edgupgren 15815

Properties of an edge of a pseudograph. (Contributed by AV, 8-Nov-2020.)

|- ( ( G e. UPGraph /\ E e. (Edg ` G ) ) -> ( E e. ~P (Vtx ` G ) /\ ( E ~~ 1o \/ E ~~ 2o ) ) )

Theorem

edgumgren 15816

Properties of an edge of a multigraph. (Contributed by AV, 25-Nov-2020.)

|- ( ( G e. UMGraph /\ E e. (Edg ` G ) ) -> ( E e. ~P (Vtx ` G ) /\ E ~~ 2o ) )

Theorem

uhgrvtxedgiedgb 15817*

In a hypergraph, a vertex is incident with an edge iff it is contained in an element of the range of the edge function. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 6-Jul-2022.)

|- I = (iEdg ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UHGraph /\ U e. V ) -> ( E. i e. dom I U e. ( I ` i ) <-> E. e e. E U e. e ) )

Theorem

upgredg 15818*

For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.) (Proof shortened by AV, 26-Nov-2021.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UPGraph /\ C e. E ) -> E. a e. V E. b e. V C = { a , b } )

Theorem

umgredg 15819*

For each edge in a multigraph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 25-Nov-2020.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UMGraph /\ C e. E ) -> E. a e. V E. b e. V ( a =/= b /\ C = { a , b } ) )

Theorem

upgrpredgv 15820

An edge of a pseudograph always connects two vertices if the edge contains two sets. The two vertices/sets need not necessarily be different (loops are allowed). (Contributed by AV, 18-Nov-2021.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UPGraph /\ ( M e. U /\ N e. W ) /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )

Theorem

umgrpredgv 15821

An edge of a multigraph always connects two vertices. This theorem does not hold for arbitrary pseudographs: if either M or N is a proper class, then { M , N } e. E could still hold ( { M , N } would be either { M } or { N }, see prprc1 3746 or prprc2 3747, i.e. a loop), but M e. V or N e. V would not be true. (Contributed by AV, 27-Nov-2020.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UMGraph /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )

Theorem

upgredg2vtx 15822*

For a vertex incident to an edge there is another vertex incident to the edge in a pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 5-Dec-2020.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( G e. UPGraph /\ C e. E /\ A e. C ) -> E. b e. V C = { A , b } )

Theorem

upgredgpr 15823

If a proper pair (of vertices) is a subset of an edge in a pseudograph, the pair is the edge. (Contributed by AV, 30-Dec-2020.)

|- V = (Vtx ` G )   &    |- E = (Edg ` G )   =>    |- ( ( ( G e. UPGraph /\ C e. E /\ { A , B } C_ C ) /\ ( A e. U /\ B e. W /\ A =/= B ) ) -> { A , B } = C )

Theorem

umgredgne 15824

An edge of a multigraph always connects two different vertices. Analogue of umgrnloopvv 15795. (Contributed by AV, 27-Nov-2020.)

|- E = (Edg ` G )   =>    |- ( ( G e. UMGraph /\ { M , N } e. E ) -> M =/= N )

Theorem

umgrnloop2 15825

A multigraph has no loops. (Contributed by AV, 27-Oct-2020.) (Revised by AV, 30-Nov-2020.)

|- ( G e. UMGraph -> { N , N } e/ (Edg ` G ) )

Theorem

umgredgnlp 15826*

An edge of a multigraph is not a loop. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 8-Jun-2021.)

|- E = (Edg ` G )   =>    |- ( ( G e. UMGraph /\ C e. E ) -> -. E. v C = { v } )

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 15827

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 15917 (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 5961, 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 5626 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 7390, df-ap 8685		Yes	apadd1 8711, apne 8726
g	with "is a set" condition			No	1stvalg 6246, brtposg 6358, setsmsbasg 15036
m	inhabited (from "member")		E. x x e. A	No	r19.2m 3551, negm 9766, ctm 7232, basmex 12976
seq3, sum3	recursive sequence	df-seqfrec 10625		Yes	seq3-1 10639, fsum3 11783
tap	tight apartness	df-tap 7392		Yes	df-tap 7392

• Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7392.

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

|- ph   =>    |- ph

13.1.2  Definitional examples

Theorem

ex-or 15828

Example for ax-io 711. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

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

Theorem

ex-an 15829

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

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

Theorem

1kp2ke3k 15830

Example for df-dec 9535, 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 9535 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 15831

Example for df-fl 10445. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

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

Theorem

ex-ceil 15832

Example for df-ceil 10446. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-exp 15833

Example for df-exp 10716. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-fac 15834

Example for df-fac 10903. (Contributed by AV, 4-Sep-2021.)

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

Theorem

ex-bc 15835

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

|- ( 5 _C 3 ) = ; 1 0

Theorem

ex-dvds 15836

Example for df-dvds 12184: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

|- 3 || 6

Theorem

ex-gcd 15837

Example for df-gcd 12360. (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 15838

(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 15839

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 15840

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

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

Theorem

bj-nnim 15841

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 15842

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 15843

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

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

Theorem

bj-imnimnn 15844

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 15843 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 15846 or bj-stfal 15848 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 15845

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

|- ( ph -> STAB ph )

Theorem

bj-sttru 15846

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

|- STAB T.

Theorem

bj-fast 15847

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

|- ( -. ph -> STAB ph )

Theorem

bj-stfal 15848

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

|- STAB F.

Theorem

bj-nnst 15849

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 16095 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 15850

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 15863). (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stst 15851

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 15852

A conjunction with a stable consequent is stable. See stabnot 835 for negation , bj-stan 15853 for conjunction , and bj-stal 15855 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stan 15853

The conjunction of two stable formulas is stable. See bj-stim 15852 for implication, stabnot 835 for negation, and bj-stal 15855 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-stand 15854

The conjunction of two stable formulas is stable. Deduction form of bj-stan 15853. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 15853 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 15855

The universal quantification of a stable formula is stable. See bj-stim 15852 for implication, stabnot 835 for negation, and bj-stan 15853 for conjunction. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-pm2.18st 15856

Clavius law for stable formulas. See pm2.18dc 857. (Contributed by BJ, 4-Dec-2023.)

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

Theorem

bj-con1st 15857

Contraposition when the antecedent is a negated stable proposition. See con1dc 858. (Contributed by BJ, 11-Nov-2024.)

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

14.2.1.2  Decidable formulas

Theorem

bj-trdc 15858

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

|- ( ph -> DECID ph )

Theorem

bj-dctru 15859

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

|- DECID T.

Theorem

bj-fadc 15860

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

|- ( -. ph -> DECID ph )

Theorem

bj-dcfal 15861

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

|- DECID F.

Theorem

bj-dcstab 15862

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

|- (DECID ph -> STAB ph )

Theorem

bj-nnbidc 15863

If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 15850. (Contributed by BJ, 24-Nov-2023.)

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

Theorem

bj-nndcALT 15864

Alternate proof of nndc 853. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)

|- -. -. DECID ph

Theorem

bj-dcdc 15865

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 15866

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 15867

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 15868*

Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1622 and 19.9ht 1665 or 19.23ht 1521). (Proof modification is discouraged.)

|- ( E. x ph -> ph )

Theorem

bj-hbalt 15869

Closed form of hbal 1501 (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 15870

Closed form of nfal 1600 (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 15871

Deduction form of spim 1762. (Contributed by BJ, 17-Oct-2019.)

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

Theorem

2spim 15872*

Double substitution, as in spim 1762. (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 15873*

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 15874*

Version of ch2var 15873 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 15875

Lemma for bj-vtoclgf 15882. (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 15876

Lemma for bj-vtoclgf 15882. (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 15877

A strengthening of sbiedh 1811 (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 15878

A strengthening of sbieh 1814 (same proof). (Contributed by BJ, 16-Dec-2019.)

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

Theorem

bj-sbime 15879

A strengthening of sbie 1815 (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 15880

Shorter proof of el2oss1o 6547 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 15881

Weakening two hypotheses of vtoclgf 2833. (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 15882

Weakening two hypotheses of vtoclgf 2833. (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 15883

Lemma for elabgf 2919. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabgft1 15884

One implication of elabgf 2919, 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 15885

One implication of elabgf 2919. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabgf2 15886

One implication of elabgf 2919. (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 15887*

One implication of elabf 2920. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabf2 15888*

One implication of elabf 2920. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elab1 15889*

One implication of elab 2921. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elab2a 15890*

One implication of elab 2921. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

elabg2 15891*

One implication of elabg 2923. (Contributed by BJ, 21-Nov-2019.)

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

Theorem

bj-rspgt 15892

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2878 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 15893

Restricted specialization, generalized. Weakens a hypothesis of rspccv 2878 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 15894*

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 15895

Version of intss1 3909 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 15896

Version of intss1 3909 using a class abstraction and implicit substitution. Closed form of intmin3 3921. (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 15897

Version of elssuni 3887 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 15898

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 15900).

Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 15946).

Syntax

wdcin 15899

Syntax for decidability of a class in another.

wff A DECID_in B

Definition

df-dcin 15900*

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 )