P. 157 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15601-15700   *Has distinct variable group(s)

Type	Label	Description
Statement
13.2.10.3  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 15601*	Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4636 for a nonconstructive proof of the general case. See bdfind 15602 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 15602*	Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4636 for a nonconstructive proof of the general case. See findset 15601 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 15603*	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 4637 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4637, finds2 4638, finds1 4639. (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 15604*	Version of bj-bdfindis 15603 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 15603 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 15605	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 15603 for explanations. From this version, it is easy to prove the bounded version of findes 4640. (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 15606*	Constructive proof of a variant of nn0suc 4641. For a constructive proof of nn0suc 4641, see bj-nn0suc 15620. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )
Theorem	bj-nntrans 15607	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 15608	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 15609	A natural number does not belong to itself. Version of elirr 4578 for natural numbers, which does not require ax-setind 4574. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> -. A e. A )
Theorem	bj-nnen2lp 15610	A version of en2lp 4591 for natural numbers, which does not require ax-setind 4574. Note: using this theorem and bj-nnelirr 15609, one can remove dependency on ax-setind 4574 from nntri2 6553 and nndcel 6559; 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 15611	Remove from peano4 4634 dependency on ax-setind 4574. 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 15612	The set om is transitive. A natural number is included in om. Constructive proof of elnn 4643. 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 15613	The set om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Tr om
Theorem	bj-nnord 15614	A natural number is an ordinal class. Constructive proof of nnord 4649. Can also be proved from bj-nnelon 15615 if the latter is proved from bj-omssonALT 15619. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> Ord A )
Theorem	bj-nnelon 15615	A natural number is an ordinal. Constructive proof of nnon 4647. Can also be proved from bj-omssonALT 15619. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> A e. On )
Theorem	bj-omord 15616	The set om is an ordinal class. Constructive proof of ordom 4644. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Ord om
Theorem	bj-omelon 15617	The set om is an ordinal. Constructive proof of omelon 4646. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- om e. On
Theorem	bj-omsson 15618	Constructive proof of omsson 4650. See also bj-omssonALT 15619. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
|- om C_ On
Theorem	bj-omssonALT 15619	Alternate proof of bj-omsson 15618. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
|- om C_ On
Theorem	bj-nn0suc 15620*	Proof of (biconditional form of) nn0suc 4641 from the core axioms of CZF. See also bj-nn0sucALT 15634. 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 ) )
13.2.11  CZF: Set induction In this section, we add the axiom of set induction to the core axioms of CZF.
13.2.11.1  Set induction In this section, we prove some variants of the axiom of set induction.
Theorem	setindft 15621*	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 15622*	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 15623*	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 15624*	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 15625*	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 15626*	Lemma for bj-inf2vn 15630. 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 15627*	Lemma for bj-inf2vnlem3 15628 and bj-inf2vnlem4 15629. 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 15628*	Lemma for bj-inf2vn 15630. (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 15629*	Lemma for bj-inf2vn2 15631. (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 15630*	A sufficient condition for om to be a set. See bj-inf2vn2 15631 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 15631*	A sufficient condition for om to be a set; unbounded version of bj-inf2vn 15630. (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 15632*	Another axiom of infinity in a constructive setting (see ax-infvn 15597). (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 15633	Using bounded set induction and the strong axiom of infinity, om is a set, that is, we recover ax-infvn 15597 (see bj-2inf 15594 for the equivalence of the latter with bj-omex 15598). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- om e. _V
Theorem	bj-nn0sucALT 15634*	Alternate proof of bj-nn0suc 15620, also constructive but from ax-inf2 15632, hence requiring ax-bdsetind 15624. (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 ) )
13.2.11.2  Full induction In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.
Theorem	bj-findis 15635*	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 15603 for a bounded version not requiring ax-setind 4574. See finds 4637 for a proof in IZF. From this version, it is easy to prove of finds 4637, finds2 4638, finds1 4639. (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 15636*	Version of bj-findis 15635 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 15635 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 15637	Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 15635 for explanations. From this version, it is easy to prove findes 4640. (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 )
13.2.12  CZF: Strong collection In this section, we state the axiom scheme of strong collection, which is part of CZF set theory.
Axiom	ax-strcoll 15638*	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 4149. (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 15639*	Version of ax-strcoll 15638 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 15640*	Closed form of strcollnf 15641. (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 15641*	Version of ax-strcoll 15638 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 15639 with the disjoint variable condition on b , ph replaced with a nonfreeness hypothesis. This proof aims to demonstrate a standard technique, but strcoll2 15639 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 15642*	Alternate proof of strcollnf 15641, not using strcollnft 15640. (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 ) )
13.2.13  CZF: Subset collection In this section, we state the axiom scheme of subset collection, which is part of CZF set theory.
Axiom	ax-sscoll 15643*	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 15644*	Version of ax-sscoll 15643 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 ) )
13.2.14  Real numbers
Axiom	ax-ddkcomp 15645	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 15645 should be used in place of construction specific results. In particular, axcaucvg 7969 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 ) ) )
13.3  Mathbox for Jim Kingdon
13.3.1  Propositional and predicate logic
Theorem	nnnotnotr 15646	Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 851, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.)
|- -. -. ( -. -. ph -> ph )
13.3.2  Natural numbers
Theorem	1dom1el 15647	If a set is dominated by one, then any two of its elements are equal. (Contributed by Jim Kingdon, 23-Apr-2025.)
|- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> B = C )
Theorem	ss1oel2o 15648	Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4232 which more directly illustrates the contrast with el2oss1o 6502. (Contributed by Jim Kingdon, 8-Aug-2022.)
|- (EXMID <-> A. x ( x C_ 1o -> x e. 2o ) )
Theorem	nnti 15649	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 15650	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 15651	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 }
13.3.3  The power set of a singleton
Theorem	pwtrufal 15652	A subset of the singleton { (/) } cannot be anything other than (/) or { (/) }. Removing the double negation would change the meaning, as seen at exmid01 4232. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4230), 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 15653*	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 15654*	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	subctctexmid 15655*	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 15656*	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 15657*	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 ) )
13.3.4  Omniscience of NN+oo
Theorem	0nninf 15658	The zero element of ℕ∞ (the constant sequence equal to (/)). (Contributed by Jim Kingdon, 14-Jul-2022.)
|- ( om X. { (/) } ) e. ℕ∞
Theorem	nnsf 15659*	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 15660*	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 15661*	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 15662*	Lemma for nninfall 15663. (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 15663*	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 15664*	Lemma for nninfself 15667. 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 15665*	Lemma for nninfself 15667. (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 15666*	Lemma for nninfself 15667. (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 15667*	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 15668*	Lemma for nninfsel 15671. (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 15669*	Lemma for nninfsel 15671. (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 15670*	Lemma for nninfsel 15671. (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 15671*	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 15672*	Lemma for nninfomni 15673. (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 15673	ℕ∞ is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.)
|- ℕ∞ e. Omni
Theorem	nninffeq 15674*	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 )
Theorem	nnnninfen 15675	Equinumerosity of the natural numbers and ℕ∞ is equivalent to the Limited Principle of Omniscience (LPO). Remark in Section 1.1 of [Pradic2025], p. 2. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- ( om ~~ ℕ∞ <-> om e. Omni )
13.3.5  Schroeder-Bernstein Theorem
Theorem	exmidsbthrlem 15676*	Lemma for exmidsbthr 15677. (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 15677*	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 15678*	The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 7034) 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 7034. The reverse direction (exmidsbthr 15677) 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 15679	Lemma for sbthom 15680. (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 15680	Schroeder-Bernstein is not possible even for om. We know by exmidsbth 15678 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 )
13.3.6  Real and complex numbers
Theorem	qdencn 15681*	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 11369 (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 15682*	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 15683	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 ) )
Theorem	isomninnlem 15684*	Lemma for isomninn 15685. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( A e. V -> ( A e. Omni <-> A. f e. ( { 0 , 1 } ^m A ) ( E. x e. A ( f ` x ) = 0 \/ A. x e. A ( f ` x ) = 1 ) ) )
Theorem	isomninn 15685*	Omniscience stated in terms of natural numbers. Similar to isomnimap 7204 but it will sometimes be more convenient to use 0 and 1 rather than (/) and 1o. (Contributed by Jim Kingdon, 30-Aug-2023.)
|- ( A e. V -> ( A e. Omni <-> A. f e. ( { 0 , 1 } ^m A ) ( E. x e. A ( f ` x ) = 0 \/ A. x e. A ( f ` x ) = 1 ) ) )
Theorem	cvgcmp2nlemabs 15686*	Lemma for cvgcmp2n 15687. The partial sums get closer to each other as we go further out. The proof proceeds by rewriting ( seq 1 ( + , G ) ` N ) as the sum of ( seq 1 ( + , G ) ` M ) and a term which gets smaller as M gets large. (Contributed by Jim Kingdon, 25-Aug-2023.)
|- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )   &    |- ( ( ph /\ k e. NN ) -> 0 <_ ( G ` k ) )   &    |- ( ( ph /\ k e. NN ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> ( abs ` ( ( seq 1 ( + , G ) ` N ) - ( seq 1 ( + , G ) ` M ) ) ) < ( 2 / M ) )
Theorem	cvgcmp2n 15687*	A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.)
|- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )   &    |- ( ( ph /\ k e. NN ) -> 0 <_ ( G ` k ) )   &    |- ( ( ph /\ k e. NN ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )   =>    |- ( ph -> seq 1 ( + , G ) e. dom ~~> )
Theorem	iooref1o 15688	A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.)
|- F = ( x e. RR |-> ( 1 / ( 1 + ( exp ` x ) ) ) )   =>    |- F : RR -1-1-onto-> ( 0 (,) 1 )
Theorem	iooreen 15689	An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)
|- ( 0 (,) 1 ) ~~ RR
13.3.7  Analytic omniscience principles Omniscience principles refer to several propositions, most of them weaker than full excluded middle, which do not follow from the axioms of IZF set theory. They are: (0) the Principle of Omniscience (PO), which is another name for excluded middle (see exmidomni 7209), (1) the Limited Principle of Omniscience (LPO) is om e. Omni (see df-omni 7202), (2) the Weak Limited Principle of Omniscience (WLPO) is om e. WOmni (see df-womni 7231), (3) Markov's Principle (MP) is om e. Markov (see df-markov 7219), (4) the Lesser Limited Principle of Omniscience (LLPO) is not yet defined in iset.mm. They also have analytic counterparts each of which follows from the corresponding omniscience principle: (1) Analytic LPO is real number trichotomy, A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) (see trilpo 15697), (2) Analytic WLPO is decidability of real number equality, A. x e. RR A. y e. RRDECID x = y (see redcwlpo 15709), (3) Analytic MP is A. x e. RR A. y e. RR ( x =/= y -> x # y ) (see neapmkv 15722), (4) Analytic LLPO is real number dichotomy, A. x e. RR A. y e. RR ( x <_ y \/ y <_ x ) (most relevant current theorem is maxclpr 11389).
Theorem	trilpolemclim 15690*	Lemma for trilpo 15697. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- G = ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) )   =>    |- ( ph -> seq 1 ( + , G ) e. dom ~~> )
Theorem	trilpolemcl 15691*	Lemma for trilpo 15697. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   =>    |- ( ph -> A e. RR )
Theorem	trilpolemisumle 15692*	Lemma for trilpo 15697. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. NN )   =>    |- ( ph -> sum_ i e. Z ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ sum_ i e. Z ( 1 / ( 2 ^ i ) ) )
Theorem	trilpolemgt1 15693*	Lemma for trilpo 15697. The 1 < A case. (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   =>    |- ( ph -> -. 1 < A )
Theorem	trilpolemeq1 15694*	Lemma for trilpo 15697. The A = 1 case. This is proved by noting that if any ( F ` x ) is zero, then the infinite sum A is less than one based on the term which is zero. We are using the fact that the F sequence is decidable (in the sense that each element is either zero or one). (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   &    |- ( ph -> A = 1 )   =>    |- ( ph -> A. x e. NN ( F ` x ) = 1 )
Theorem	trilpolemlt1 15695*	Lemma for trilpo 15697. The A < 1 case. We can use the distance between A and one (that is, 1 - A) to find a position in the sequence n where terms after that point will not add up to as much as 1 - A. By finomni 7207 we know the terms up to n either contain a zero or are all one. But if they are all one that contradicts the way we constructed n, so we know that the sequence contains a zero. (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   &    |- ( ph -> A < 1 )   =>    |- ( ph -> E. x e. NN ( F ` x ) = 0 )
Theorem	trilpolemres 15696*	Lemma for trilpo 15697. The result. (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   &    |- ( ph -> ( A < 1 \/ A = 1 \/ 1 < A ) )   =>    |- ( ph -> ( E. x e. NN ( F ` x ) = 0 \/ A. x e. NN ( F ` x ) = 1 ) )
Theorem	trilpo 15697*	Real number trichotomy implies the Limited Principle of Omniscience (LPO). We expect that we'd need some form of countable choice to prove the converse. Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence contains a zero or it is all ones. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. Compare it with one using trichotomy. The three cases from trichotomy are trilpolemlt1 15695 (which means the sequence contains a zero), trilpolemeq1 15694 (which means the sequence is all ones), and trilpolemgt1 15693 (which is not possible). Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 15683) or that the real numbers are a discrete field (see trirec0 15698). LPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qtri3or 10332 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) -> om e. Omni )
Theorem	trirec0 15698*	Every real number having a reciprocal or equaling zero is equivalent to real number trichotomy. This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 15697). (Contributed by Jim Kingdon, 10-Jun-2024.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. x e. RR ( E. z e. RR ( x x. z ) = 1 \/ x = 0 ) )
Theorem	trirec0xor 15699*	Version of trirec0 15698 with exclusive-or. The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. x e. RR ( E. z e. RR ( x x. z ) = 1 \/_ x = 0 ) )
Theorem	apdifflemf 15700	Lemma for apdiff 15702. Being apart from the point halfway between Q and R suffices for A to be a different distance from Q and from R. (Contributed by Jim Kingdon, 18-May-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> Q e. QQ )   &    |- ( ph -> R e. QQ )   &    |- ( ph -> Q < R )   &    |- ( ph -> ( ( Q + R ) / 2 ) # A )   =>    |- ( ph -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )