P. 136 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13501-13600   *Has distinct variable group(s)

Type	Label	Description
Statement
11.2.9.2  Inductive classes and the class of natural number ordinals
Syntax	wind 13501	Syntax for inductive classes.
wff Ind A
Definition	df-bj-ind 13502*	Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
|- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
Theorem	bj-indsuc 13503	A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
|- (Ind A -> ( B e. A -> suc B e. A ) )
Theorem	bj-indeq 13504	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
|- ( A = B -> (Ind A <-> Ind B ) )
Theorem	bj-bdind 13505	Boundedness of the formula "the setvar x is an inductive class". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED Ind x
Theorem	bj-indint 13506*	The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
|- Ind |^| { x e. A | Ind x }
Theorem	bj-indind 13507*	If A is inductive and B is "inductive in A", then ( A i^i B ) is inductive. (Contributed by BJ, 25-Oct-2020.)
|- ( (Ind A /\ ( (/) e. B /\ A. x e. A ( x e. B -> suc x e. B ) ) ) -> Ind ( A i^i B ) )
Theorem	bj-dfom 13508	Alternate definition of om, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.)
|- om = |^| { x | Ind x }
Theorem	bj-omind 13509	om is an inductive class. (Contributed by BJ, 30-Nov-2019.)
|- Ind om
Theorem	bj-omssind 13510	om is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> (Ind A -> om C_ A ) )
Theorem	bj-ssom 13511*	A characterization of subclasses of om. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( A. x (Ind x -> A C_ x ) <-> A C_ om )
Theorem	bj-om 13512*	A set is equal to om if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> ( A = om <-> (Ind A /\ A. x (Ind x -> A C_ x ) ) ) )
Theorem	bj-2inf 13513*	Two formulations of the axiom of infinity (see ax-infvn 13516 and bj-omex 13517) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( om e. _V <-> E. x (Ind x /\ A. y (Ind y -> x C_ y ) ) )
11.2.9.3  The first three Peano postulates The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4553 and peano3 4555 already show this. In this section, we prove bj-peano2 13514 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.
Theorem	bj-peano2 13514	Constructive proof of peano2 4554. Temporary note: another possibility is to simply replace sucexg 4457 with bj-sucexg 13497 in the proof of peano2 4554. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> suc A e. om )
Theorem	peano5set 13515*	Version of peano5 4557 when om i^i A is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( ( om i^i A ) e. V -> ( ( (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A ) )
11.2.10  CZF: Infinity In the absence of full separation, the axiom of infinity has to be stated more precisely, as the existence of the smallest class containing the empty set and the successor of each of its elements.
11.2.10.1  The set of natural number ordinals In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 13516) and deduce that the class om of natural number ordinals is a set (bj-omex 13517).
Axiom	ax-infvn 13516*	Axiom of infinity in a constructive setting. This asserts the existence of the special set we want (the set of natural numbers), instead of the existence of a set with some properties (ax-iinf 4547) from which one then proves, using full separation, that the wanted set exists (omex 4552). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
|- E. x (Ind x /\ A. y (Ind y -> x C_ y ) )
Theorem	bj-omex 13517	Proof of omex 4552 from ax-infvn 13516. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
|- om e. _V
11.2.10.2  Peano's fifth postulate In this section, we give constructive proofs of two versions of Peano's fifth postulate.
Theorem	bdpeano5 13518*	Bounded version of peano5 4557. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( ( (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A )
Theorem	speano5 13519*	Version of peano5 4557 when A is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( ( A e. V /\ (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A )
11.2.10.3  Bounded induction and Peano's fourth postulate In this section, we prove various versions of bounded induction from the basic axioms of CZF (in particular, without the axiom of set induction). We also prove Peano's fourth postulate. Together with the results from the previous sections, this proves from the core axioms of CZF (with infinity) that the set of natural number ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers.
Theorem	findset 13520*	Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4558 for a nonconstructive proof of the general case. See bdfind 13521 for a proof when A is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A = om ) )
Theorem	bdfind 13521*	Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4558 for a nonconstructive proof of the general case. See findset 13520 for a proof when A is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A = om )
Theorem	bj-bdfindis 13522*	Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4559 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4559, finds2 4560, finds1 4561. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- F/ x ps   &    |- F/ x ch   &    |- F/ x th   &    |- ( x = (/) -> ( ps -> ph ) )   &    |- ( x = y -> ( ph -> ch ) )   &    |- ( x = suc y -> ( th -> ph ) )   =>    |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )
Theorem	bj-bdfindisg 13523*	Version of bj-bdfindis 13522 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13522 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- F/ x ps   &    |- F/ x ch   &    |- F/ x th   &    |- ( x = (/) -> ( ps -> ph ) )   &    |- ( x = y -> ( ph -> ch ) )   &    |- ( x = suc y -> ( th -> ph ) )   &    |- F/_ x A   &    |- F/ x ta   &    |- ( x = A -> ( ph -> ta ) )   =>    |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> ( A e. om -> ta ) )
Theorem	bj-bdfindes 13524	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13522 for explanations. From this version, it is easy to prove the bounded version of findes 4562. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   =>    |- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )
Theorem	bj-nn0suc0 13525*	Constructive proof of a variant of nn0suc 4563. For a constructive proof of nn0suc 4563, see bj-nn0suc 13539. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )
Theorem	bj-nntrans 13526	A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> ( B e. A -> B C_ A ) )
Theorem	bj-nntrans2 13527	A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> Tr A )
Theorem	bj-nnelirr 13528	A natural number does not belong to itself. Version of elirr 4500 for natural numbers, which does not require ax-setind 4496. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> -. A e. A )
Theorem	bj-nnen2lp 13529	A version of en2lp 4513 for natural numbers, which does not require ax-setind 4496. Note: using this theorem and bj-nnelirr 13528, one can remove dependency on ax-setind 4496 from nntri2 6441 and nndcel 6447; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)
|- ( ( A e. om /\ B e. om ) -> -. ( A e. B /\ B e. A ) )
Theorem	bj-peano4 13530	Remove from peano4 4556 dependency on ax-setind 4496. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)
|- ( ( A e. om /\ B e. om ) -> ( suc A = suc B <-> A = B ) )
Theorem	bj-omtrans 13531	The set om is transitive. A natural number is included in om. Constructive proof of elnn 4565. The idea is to use bounded induction with the formula x C_ om. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with x C_ a and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- ( A e. om -> A C_ om )
Theorem	bj-omtrans2 13532	The set om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Tr om
Theorem	bj-nnord 13533	A natural number is an ordinal class. Constructive proof of nnord 4571. Can also be proved from bj-nnelon 13534 if the latter is proved from bj-omssonALT 13538. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> Ord A )
Theorem	bj-nnelon 13534	A natural number is an ordinal. Constructive proof of nnon 4569. Can also be proved from bj-omssonALT 13538. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> A e. On )
Theorem	bj-omord 13535	The set om is an ordinal class. Constructive proof of ordom 4566. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Ord om
Theorem	bj-omelon 13536	The set om is an ordinal. Constructive proof of omelon 4568. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- om e. On
Theorem	bj-omsson 13537	Constructive proof of omsson 4572. See also bj-omssonALT 13538. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
|- om C_ On
Theorem	bj-omssonALT 13538	Alternate proof of bj-omsson 13537. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
|- om C_ On
Theorem	bj-nn0suc 13539*	Proof of (biconditional form of) nn0suc 4563 from the core axioms of CZF. See also bj-nn0sucALT 13553. As a characterization of the elements of om, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om <-> ( A = (/) \/ E. x e. om A = suc x ) )
11.2.11  CZF: Set induction In this section, we add the axiom of set induction to the core axioms of CZF.
11.2.11.1  Set induction In this section, we prove some variants of the axiom of set induction.
Theorem	setindft 13540*	Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
|- ( A. x F/ y ph -> ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph ) )
Theorem	setindf 13541*	Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
|- F/ y ph   =>    |- ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph )
Theorem	setindis 13542*	Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
|- F/ x ps   &    |- F/ x ch   &    |- F/ y ph   &    |- F/ y ps   &    |- ( x = z -> ( ph -> ps ) )   &    |- ( x = y -> ( ch -> ph ) )   =>    |- ( A. y ( A. z e. y ps -> ch ) -> A. x ph )
Axiom	ax-bdsetind 13543*	Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
|- BOUNDED ph   =>    |- ( A. a ( A. y e. a [ y / a ] ph -> ph ) -> A. a ph )
Theorem	bdsetindis 13544*	Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- F/ x ps   &    |- F/ x ch   &    |- F/ y ph   &    |- F/ y ps   &    |- ( x = z -> ( ph -> ps ) )   &    |- ( x = y -> ( ch -> ph ) )   =>    |- ( A. y ( A. z e. y ps -> ch ) -> A. x ph )
Theorem	bj-inf2vnlem1 13545*	Lemma for bj-inf2vn 13549. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> Ind A )
Theorem	bj-inf2vnlem2 13546*	Lemma for bj-inf2vnlem3 13547 and bj-inf2vnlem4 13548. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A. u ( A. t e. u ( t e. A -> t e. Z ) -> ( u e. A -> u e. Z ) ) ) )
Theorem	bj-inf2vnlem3 13547*	Lemma for bj-inf2vn 13549. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- BOUNDED Z   =>    |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )
Theorem	bj-inf2vnlem4 13548*	Lemma for bj-inf2vn2 13550. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )
Theorem	bj-inf2vn 13549*	A sufficient condition for om to be a set. See bj-inf2vn2 13550 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( A e. V -> ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A = om ) )
Theorem	bj-inf2vn2 13550*	A sufficient condition for om to be a set; unbounded version of bj-inf2vn 13549. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- ( A e. V -> ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A = om ) )
Axiom	ax-inf2 13551*	Another axiom of infinity in a constructive setting (see ax-infvn 13516). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
|- E. a A. x ( x e. a <-> ( x = (/) \/ E. y e. a x = suc y ) )
Theorem	bj-omex2 13552	Using bounded set induction and the strong axiom of infinity, om is a set, that is, we recover ax-infvn 13516 (see bj-2inf 13513 for the equivalence of the latter with bj-omex 13517). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- om e. _V
Theorem	bj-nn0sucALT 13553*	Alternate proof of bj-nn0suc 13539, also constructive but from ax-inf2 13551, hence requiring ax-bdsetind 13543. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A e. om <-> ( A = (/) \/ E. x e. om A = suc x ) )
11.2.11.2  Full induction In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.
Theorem	bj-findis 13554*	Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 13522 for a bounded version not requiring ax-setind 4496. See finds 4559 for a proof in IZF. From this version, it is easy to prove of finds 4559, finds2 4560, finds1 4561. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.)
|- F/ x ps   &    |- F/ x ch   &    |- F/ x th   &    |- ( x = (/) -> ( ps -> ph ) )   &    |- ( x = y -> ( ph -> ch ) )   &    |- ( x = suc y -> ( th -> ph ) )   =>    |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )
Theorem	bj-findisg 13555*	Version of bj-findis 13554 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 13554 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- F/ x ps   &    |- F/ x ch   &    |- F/ x th   &    |- ( x = (/) -> ( ps -> ph ) )   &    |- ( x = y -> ( ph -> ch ) )   &    |- ( x = suc y -> ( th -> ph ) )   &    |- F/_ x A   &    |- F/ x ta   &    |- ( x = A -> ( ph -> ta ) )   =>    |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> ( A e. om -> ta ) )
Theorem	bj-findes 13556	Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 13554 for explanations. From this version, it is easy to prove findes 4562. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )
11.2.12  CZF: Strong collection In this section, we state the axiom scheme of strong collection, which is part of CZF set theory.
Axiom	ax-strcoll 13557*	Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that ph represents a multivalued function on a, or equivalently a collection of nonempty classes indexed by a, and the axiom asserts the existence of a set b which "collects" at least one element in the image of each x e. a and which is made only of such elements. That second conjunct is what makes it "strong", compared to the axiom scheme of collection ax-coll 4079. (Contributed by BJ, 5-Oct-2019.)
|- A. a ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
Theorem	strcoll2 13558*	Version of ax-strcoll 13557 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
|- ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
Theorem	strcollnft 13559*	Closed form of strcollnf 13560. (Contributed by BJ, 21-Oct-2019.)
|- ( A. x A. y F/ b ph -> ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) ) )
Theorem	strcollnf 13560*	Version of ax-strcoll 13557 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 13558 with the disjoint variable condition on b , ph replaced with a nonfreeness hypothesis. This proof aims to demonstrate a standard technique, but strcoll2 13558 will generally suffice: since the theorem asserts the existence of a set b, supposing that that setvar does not occur in the already defined ph is not a big constraint. (Contributed by BJ, 21-Oct-2019.)
|- F/ b ph   =>    |- ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
Theorem	strcollnfALT 13561*	Alternate proof of strcollnf 13560, not using strcollnft 13559. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ b ph   =>    |- ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
11.2.13  CZF: Subset collection In this section, we state the axiom scheme of subset collection, which is part of CZF set theory.
Axiom	ax-sscoll 13562*	Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that ph represents a multivalued function from a to b, or equivalently a collection of nonempty subsets of b indexed by a, and the consequent asserts the existence of a subset of c which "collects" at least one element in the image of each x e. a and which is made only of such elements. The axiom asserts the existence, for any sets a , b, of a set c such that that implication holds for any value of the parameter z of ph. (Contributed by BJ, 5-Oct-2019.)
|- A. a A. b E. c A. z ( A. x e. a E. y e. b ph -> E. d e. c ( A. x e. a E. y e. d ph /\ A. y e. d E. x e. a ph ) )
Theorem	sscoll2 13563*	Version of ax-sscoll 13562 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
|- E. c A. z ( A. x e. a E. y e. b ph -> E. d e. c ( A. x e. a E. y e. d ph /\ A. y e. d E. x e. a ph ) )
11.2.14  Real numbers
Axiom	ax-ddkcomp 13564	Axiom of Dedekind completeness for Dedekind real numbers: every inhabited upper-bounded located set of reals has a real upper bound. Ideally, this axiom should be "proved" as "axddkcomp" for the real numbers constructed from IZF, and then Axiom ax-ddkcomp 13564 should be used in place of construction specific results. In particular, axcaucvg 7820 should be proved from it. (Contributed by BJ, 24-Oct-2021.)
|- ( ( ( A C_ RR /\ E. x x e. A ) /\ E. x e. RR A. y e. A y < x /\ A. x e. RR A. y e. RR ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) ) -> E. x e. RR ( A. y e. A y <_ x /\ ( ( B e. R /\ A. y e. A y <_ B ) -> x <_ B ) ) )
11.3  Mathbox for Jim Kingdon
11.3.1  Natural numbers
Theorem	ss1oel2o 13565	Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4159 which more directly illustrates the contrast with el2oss1o 6390. (Contributed by Jim Kingdon, 8-Aug-2022.)
|- (EXMID <-> A. x ( x C_ 1o -> x e. 2o ) )
Theorem	nnti 13566	Ordering on a natural number generates a tight apartness. (Contributed by Jim Kingdon, 7-Aug-2022.)
|- ( ph -> A e. om )   =>    |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u _E v /\ -. v _E u ) ) )
Theorem	012of 13567	Mapping zero and one between NN0 and om style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( `' G |` { 0 , 1 } ) : { 0 , 1 } --> 2o
Theorem	2o01f 13568	Mapping zero and one between om and NN0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( G |` 2o ) : 2o --> { 0 , 1 }
11.3.2  The power set of a singleton
Theorem	pwtrufal 13569	A subset of the singleton { (/) } cannot be anything other than (/) or { (/) }. Removing the double negation would change the meaning, as seen at exmid01 4159. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4157), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.)
|- ( A C_ { (/) } -> -. -. ( A = (/) \/ A = { (/) } ) )
Theorem	pwle2 13570*	An exercise related to N copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
|- T = U_ x e. N ( { x } X. 1o )   =>    |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> N C_ 2o )
Theorem	pwf1oexmid 13571*	An exercise related to N copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
|- T = U_ x e. N ( { x } X. 1o )   =>    |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> ( ran G = ~P 1o <-> ( N = 2o /\ EXMID ) ) )
Theorem	exmid1stab 13572*	If any proposition is stable, excluded middle follows. We are thinking of x as a proposition and x = { (/) } as "x is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
|- ( ( ph /\ x C_ { (/) } ) -> STAB x = { (/) } )   =>    |- ( ph -> EXMID )
Theorem	subctctexmid 13573*	If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.)
|- ( ph -> A. x ( E. s ( s C_ om /\ E. f f : s -onto-> x ) -> E. g g : om -onto-> ( x ⊔ 1o ) ) )   &    |- ( ph -> om e. Markov )   =>    |- ( ph -> EXMID )
Theorem	sssneq 13574*	Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.)
|- ( A C_ { B } -> A. y e. A A. z e. A y = z )
Theorem	pw1nct 13575*	A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.)
|- ( A. r ( r C_ ( ~P 1o X. om ) -> ( A. p e. ~P 1o E. n e. om p r n -> E. m e. om A. q e. ~P 1o q r m ) ) -> -. E. f f : om -onto-> ( ~P 1o ⊔ 1o ) )
11.3.3  Omniscience of NN+oo
Theorem	0nninf 13576	The zero element of ℕ∞ (the constant sequence equal to (/)). (Contributed by Jim Kingdon, 14-Jul-2022.)
|- ( om X. { (/) } ) e. ℕ∞
Theorem	nnsf 13577*	Domain and range of S. Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.)
|- S = ( p e. ℕ∞ |-> ( i e. om |-> if ( i = (/) , 1o , ( p ` U. i ) ) ) )   =>    |- S :ℕ∞ -->ℕ∞
Theorem	peano4nninf 13578*	The successor function on ℕ∞ is one to one. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- S = ( p e. ℕ∞ |-> ( i e. om |-> if ( i = (/) , 1o , ( p ` U. i ) ) ) )   =>    |- S :ℕ∞ -1-1->ℕ∞
Theorem	peano3nninf 13579*	The successor function on ℕ∞ is never zero. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
|- S = ( p e. ℕ∞ |-> ( i e. om |-> if ( i = (/) , 1o , ( p ` U. i ) ) ) )   =>    |- ( A e. ℕ∞ -> ( S ` A ) =/= ( x e. om |-> (/) ) )
Theorem	nninfalllem1 13580*	Lemma for nninfall 13581. (Contributed by Jim Kingdon, 1-Aug-2022.)
|- ( ph -> Q e. ( 2o ^m ℕ∞ ) )   &    |- ( ph -> ( Q ` ( x e. om |-> 1o ) ) = 1o )   &    |- ( ph -> A. n e. om ( Q ` ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) = 1o )   &    |- ( ph -> P e. ℕ∞ )   &    |- ( ph -> ( Q ` P ) = (/) )   =>    |- ( ph -> A. n e. om ( P ` n ) = 1o )
Theorem	nninfall 13581*	Given a decidable predicate on ℕ∞, showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which Q is a decidable predicate is that it assigns a value of either (/) or 1o (which can be thought of as false and true) to every element of ℕ∞. Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
|- ( ph -> Q e. ( 2o ^m ℕ∞ ) )   &    |- ( ph -> ( Q ` ( x e. om |-> 1o ) ) = 1o )   &    |- ( ph -> A. n e. om ( Q ` ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) = 1o )   =>    |- ( ph -> A. p e. ℕ∞ ( Q ` p ) = 1o )
Theorem	nninfsellemdc 13582*	Lemma for nninfself 13585. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.)
|- ( ( Q e. ( 2o ^m ℕ∞ ) /\ N e. om ) -> DECID A. k e. suc N ( Q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o )
Theorem	nninfsellemcl 13583*	Lemma for nninfself 13585. (Contributed by Jim Kingdon, 8-Aug-2022.)
|- ( ( Q e. ( 2o ^m ℕ∞ ) /\ N e. om ) -> if ( A. k e. suc N ( Q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) e. 2o )
Theorem	nninfsellemsuc 13584*	Lemma for nninfself 13585. (Contributed by Jim Kingdon, 6-Aug-2022.)
|- ( ( Q e. ( 2o ^m ℕ∞ ) /\ J e. om ) -> if ( A. k e. suc suc J ( Q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) C_ if ( A. k e. suc J ( Q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) )
Theorem	nninfself 13585*	Domain and range of the selection function for ℕ∞. (Contributed by Jim Kingdon, 6-Aug-2022.)
|- E = ( q e. ( 2o ^m ℕ∞ ) |-> ( n e. om |-> if ( A. k e. suc n ( q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) ) )   =>    |- E : ( 2o ^m ℕ∞ ) -->ℕ∞
Theorem	nninfsellemeq 13586*	Lemma for nninfsel 13589. (Contributed by Jim Kingdon, 9-Aug-2022.)
|- E = ( q e. ( 2o ^m ℕ∞ ) |-> ( n e. om |-> if ( A. k e. suc n ( q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) ) )   &    |- ( ph -> Q e. ( 2o ^m ℕ∞ ) )   &    |- ( ph -> ( Q ` ( E ` Q ) ) = 1o )   &    |- ( ph -> N e. om )   &    |- ( ph -> A. k e. N ( Q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o )   &    |- ( ph -> ( Q ` ( i e. om |-> if ( i e. N , 1o , (/) ) ) ) = (/) )   =>    |- ( ph -> ( E ` Q ) = ( i e. om |-> if ( i e. N , 1o , (/) ) ) )
Theorem	nninfsellemqall 13587*	Lemma for nninfsel 13589. (Contributed by Jim Kingdon, 9-Aug-2022.)
|- E = ( q e. ( 2o ^m ℕ∞ ) |-> ( n e. om |-> if ( A. k e. suc n ( q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) ) )   &    |- ( ph -> Q e. ( 2o ^m ℕ∞ ) )   &    |- ( ph -> ( Q ` ( E ` Q ) ) = 1o )   &    |- ( ph -> N e. om )   =>    |- ( ph -> ( Q ` ( i e. om |-> if ( i e. N , 1o , (/) ) ) ) = 1o )
Theorem	nninfsellemeqinf 13588*	Lemma for nninfsel 13589. (Contributed by Jim Kingdon, 9-Aug-2022.)
|- E = ( q e. ( 2o ^m ℕ∞ ) |-> ( n e. om |-> if ( A. k e. suc n ( q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) ) )   &    |- ( ph -> Q e. ( 2o ^m ℕ∞ ) )   &    |- ( ph -> ( Q ` ( E ` Q ) ) = 1o )   =>    |- ( ph -> ( E ` Q ) = ( i e. om |-> 1o ) )
Theorem	nninfsel 13589*	E is a selection function for ℕ∞. Theorem 3.6 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 9-Aug-2022.)
|- E = ( q e. ( 2o ^m ℕ∞ ) |-> ( n e. om |-> if ( A. k e. suc n ( q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) ) )   &    |- ( ph -> Q e. ( 2o ^m ℕ∞ ) )   &    |- ( ph -> ( Q ` ( E ` Q ) ) = 1o )   =>    |- ( ph -> A. p e. ℕ∞ ( Q ` p ) = 1o )
Theorem	nninfomnilem 13590*	Lemma for nninfomni 13591. (Contributed by Jim Kingdon, 10-Aug-2022.)
|- E = ( q e. ( 2o ^m ℕ∞ ) |-> ( n e. om |-> if ( A. k e. suc n ( q ` ( i e. om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) ) )   =>    |- ℕ∞ e. Omni
Theorem	nninfomni 13591	ℕ∞ is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.)
|- ℕ∞ e. Omni
Theorem	nninffeq 13592*	Equality of two functions on ℕ∞ which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one, |- ( ph -> A. n e. suc om ... ). (Contributed by Jim Kingdon, 4-Aug-2023.)
|- ( ph -> F :ℕ∞ --> NN0 )   &    |- ( ph -> G :ℕ∞ --> NN0 )   &    |- ( ph -> ( F ` ( x e. om |-> 1o ) ) = ( G ` ( x e. om |-> 1o ) ) )   &    |- ( ph -> A. n e. om ( F ` ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) = ( G ` ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) )   =>    |- ( ph -> F = G )
11.3.4  Schroeder-Bernstein Theorem
Theorem	exmidsbthrlem 13593*	Lemma for exmidsbthr 13594. (Contributed by Jim Kingdon, 11-Aug-2022.)
|- S = ( p e. ℕ∞ |-> ( i e. om |-> if ( i = (/) , 1o , ( p ` U. i ) ) ) )   =>    |- ( A. x A. y ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) -> EXMID )
Theorem	exmidsbthr 13594*	The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.)
|- ( A. x A. y ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) -> EXMID )
Theorem	exmidsbth 13595*	The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 6911) is the proof of the Schroeder-Bernstein Theorem from the Metamath Proof Explorer database (in which excluded middle holds), but adapted to use EXMID as an antecedent rather than being unconditionally true, as in the non-intuitionistic proof at https://us.metamath.org/mpeuni/sbth.html 6911. The reverse direction (exmidsbthr 13594) is the one which establishes that Schroeder-Bernstein implies excluded middle. This resolves the question of whether we will be able to prove Schroeder-Bernstein from our axioms in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.)
|- (EXMID <-> A. x A. y ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) )
Theorem	sbthomlem 13596	Lemma for sbthom 13597. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.)
|- ( ph -> om e. Omni )   &    |- ( ph -> Y C_ { (/) } )   &    |- ( ph -> F : om -1-1-onto-> ( Y ⊔ om ) )   =>    |- ( ph -> ( Y = (/) \/ Y = { (/) } ) )
Theorem	sbthom 13597	Schroeder-Bernstein is not possible even for om. We know by exmidsbth 13595 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is om? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.)
|- ( ( A. x ( ( x ~<_ om /\ om ~<_ x ) -> x ~~ om ) /\ om e. Omni ) -> EXMID )
11.3.5  Real and complex numbers
Theorem	qdencn 13598*	The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 11102 (and also would hold for RR X. RR with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.)
|- Q = { z e. CC | ( ( Re ` z ) e. QQ /\ ( Im ` z ) e. QQ ) }   =>    |- ( ( A e. CC /\ B e. RR+ ) -> E. x e. Q ( abs ` ( x - A ) ) < B )
Theorem	refeq 13599*	Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.)
|- ( ph -> F : RR --> RR )   &    |- ( ph -> G : RR --> RR )   &    |- ( ph -> A. x e. RR ( x < 0 -> ( F ` x ) = ( G ` x ) ) )   &    |- ( ph -> A. x e. RR ( 0 < x -> ( F ` x ) = ( G ` x ) ) )   &    |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )   =>    |- ( ph -> F = G )
Theorem	triap 13600	Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ A = B \/ B < A ) <-> DECID A # B ) )