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
Theorem	bj-peano4 15601	Remove from peano4 4633 dependency on ax-setind 4573. 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 15602	The set om is transitive. A natural number is included in om. Constructive proof of elnn 4642. 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 15603	The set om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Tr om
Theorem	bj-nnord 15604	A natural number is an ordinal class. Constructive proof of nnord 4648. Can also be proved from bj-nnelon 15605 if the latter is proved from bj-omssonALT 15609. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> Ord A )
Theorem	bj-nnelon 15605	A natural number is an ordinal. Constructive proof of nnon 4646. Can also be proved from bj-omssonALT 15609. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> A e. On )
Theorem	bj-omord 15606	The set om is an ordinal class. Constructive proof of ordom 4643. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Ord om
Theorem	bj-omelon 15607	The set om is an ordinal. Constructive proof of omelon 4645. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- om e. On
Theorem	bj-omsson 15608	Constructive proof of omsson 4649. See also bj-omssonALT 15609. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
|- om C_ On
Theorem	bj-omssonALT 15609	Alternate proof of bj-omsson 15608. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
|- om C_ On
Theorem	bj-nn0suc 15610*	Proof of (biconditional form of) nn0suc 4640 from the core axioms of CZF. See also bj-nn0sucALT 15624. 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 15611*	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 15612*	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 15613*	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 15614*	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 15615*	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 15616*	Lemma for bj-inf2vn 15620. 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 15617*	Lemma for bj-inf2vnlem3 15618 and bj-inf2vnlem4 15619. 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 15618*	Lemma for bj-inf2vn 15620. (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 15619*	Lemma for bj-inf2vn2 15621. (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 15620*	A sufficient condition for om to be a set. See bj-inf2vn2 15621 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 15621*	A sufficient condition for om to be a set; unbounded version of bj-inf2vn 15620. (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 15622*	Another axiom of infinity in a constructive setting (see ax-infvn 15587). (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 15623	Using bounded set induction and the strong axiom of infinity, om is a set, that is, we recover ax-infvn 15587 (see bj-2inf 15584 for the equivalence of the latter with bj-omex 15588). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- om e. _V
Theorem	bj-nn0sucALT 15624*	Alternate proof of bj-nn0suc 15610, also constructive but from ax-inf2 15622, hence requiring ax-bdsetind 15614. (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 15625*	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 15593 for a bounded version not requiring ax-setind 4573. See finds 4636 for a proof in IZF. From this version, it is easy to prove of finds 4636, finds2 4637, finds1 4638. (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 15626*	Version of bj-findis 15625 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 15625 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 15627	Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 15625 for explanations. From this version, it is easy to prove findes 4639. (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 15628*	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 4148. (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 15629*	Version of ax-strcoll 15628 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 15630*	Closed form of strcollnf 15631. (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 15631*	Version of ax-strcoll 15628 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 15629 with the disjoint variable condition on b , ph replaced with a nonfreeness hypothesis. This proof aims to demonstrate a standard technique, but strcoll2 15629 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 15632*	Alternate proof of strcollnf 15631, not using strcollnft 15630. (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 15633*	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 15634*	Version of ax-sscoll 15633 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 15635	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 15635 should be used in place of construction specific results. In particular, axcaucvg 7967 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 15636	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 15637	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 15638	Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4231 which more directly illustrates the contrast with el2oss1o 6501. (Contributed by Jim Kingdon, 8-Aug-2022.)
|- (EXMID <-> A. x ( x C_ 1o -> x e. 2o ) )
Theorem	nnti 15639	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 15640	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 15641	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 15642	A subset of the singleton { (/) } cannot be anything other than (/) or { (/) }. Removing the double negation would change the meaning, as seen at exmid01 4231. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4229), 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 15643*	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 15644*	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 15645*	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 15646*	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 15647*	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 15648	The zero element of ℕ∞ (the constant sequence equal to (/)). (Contributed by Jim Kingdon, 14-Jul-2022.)
|- ( om X. { (/) } ) e. ℕ∞
Theorem	nnsf 15649*	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 15650*	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 15651*	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 15652*	Lemma for nninfall 15653. (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 15653*	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 15654*	Lemma for nninfself 15657. 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 15655*	Lemma for nninfself 15657. (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 15656*	Lemma for nninfself 15657. (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 15657*	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 15658*	Lemma for nninfsel 15661. (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 15659*	Lemma for nninfsel 15661. (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 15660*	Lemma for nninfsel 15661. (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 15661*	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 15662*	Lemma for nninfomni 15663. (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 15663	ℕ∞ is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.)
|- ℕ∞ e. Omni
Theorem	nninffeq 15664*	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 15665	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 15666*	Lemma for exmidsbthr 15667. (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 15667*	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 15668*	The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 7033) 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 7033. The reverse direction (exmidsbthr 15667) 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 15669	Lemma for sbthom 15670. (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 15670	Schroeder-Bernstein is not possible even for om. We know by exmidsbth 15668 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 15671*	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 11367 (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 15672*	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 15673	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 15674*	Lemma for isomninn 15675. 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 15675*	Omniscience stated in terms of natural numbers. Similar to isomnimap 7203 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 15676*	Lemma for cvgcmp2n 15677. 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 15677*	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 15678	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 15679	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 7208), (1) the Limited Principle of Omniscience (LPO) is om e. Omni (see df-omni 7201), (2) the Weak Limited Principle of Omniscience (WLPO) is om e. WOmni (see df-womni 7230), (3) Markov's Principle (MP) is om e. Markov (see df-markov 7218), (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 15687), (2) Analytic WLPO is decidability of real number equality, A. x e. RR A. y e. RRDECID x = y (see redcwlpo 15699), (3) Analytic MP is A. x e. RR A. y e. RR ( x =/= y -> x # y ) (see neapmkv 15712), (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 11387).
Theorem	trilpolemclim 15680*	Lemma for trilpo 15687. 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 15681*	Lemma for trilpo 15687. 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 15682*	Lemma for trilpo 15687. 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 15683*	Lemma for trilpo 15687. 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 15684*	Lemma for trilpo 15687. 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 15685*	Lemma for trilpo 15687. 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 7206 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 15686*	Lemma for trilpo 15687. 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 15687*	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 15685 (which means the sequence contains a zero), trilpolemeq1 15684 (which means the sequence is all ones), and trilpolemgt1 15683 (which is not possible). Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 15673) or that the real numbers are a discrete field (see trirec0 15688). 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 10330 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 15688*	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 15687). (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 15689*	Version of trirec0 15688 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 15690	Lemma for apdiff 15692. 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 ) ) )
Theorem	apdifflemr 15691	Lemma for apdiff 15692. (Contributed by Jim Kingdon, 19-May-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> S e. QQ )   &    |- ( ph -> ( abs ` ( A - -u 1 ) ) # ( abs ` ( A - 1 ) ) )   &    |- ( ( ph /\ S =/= 0 ) -> ( abs ` ( A - 0 ) ) # ( abs ` ( A - ( 2 x. S ) ) ) )   =>    |- ( ph -> A # S )
Theorem	apdiff 15692*	The irrationals (reals apart from any rational) are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 17-May-2024.)
|- ( A e. RR -> ( A. q e. QQ A # q <-> A. q e. QQ A. r e. QQ ( q =/= r -> ( abs ` ( A - q ) ) # ( abs ` ( A - r ) ) ) ) )
Theorem	iswomninnlem 15693*	Lemma for iswomnimap 7232. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( A e. V -> ( A e. WOmni <-> A. f e. ( { 0 , 1 } ^m A )DECID A. x e. A ( f ` x ) = 1 ) )
Theorem	iswomninn 15694*	Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7232 but it will sometimes be more convenient to use 0 and 1 rather than (/) and 1o. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A e. V -> ( A e. WOmni <-> A. f e. ( { 0 , 1 } ^m A )DECID A. x e. A ( f ` x ) = 1 ) )
Theorem	iswomni0 15695*	Weak omniscience stated in terms of equality with 0. Like iswomninn 15694 but with zero in place of one. (Contributed by Jim Kingdon, 24-Jul-2024.)
|- ( A e. V -> ( A e. WOmni <-> A. f e. ( { 0 , 1 } ^m A )DECID A. x e. A ( f ` x ) = 0 ) )
Theorem	ismkvnnlem 15696*	Lemma for ismkvnn 15697. The result, with a hypothesis to give a name to an expression for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( A e. V -> ( A e. Markov <-> A. f e. ( { 0 , 1 } ^m A ) ( -. A. x e. A ( f ` x ) = 1 -> E. x e. A ( f ` x ) = 0 ) ) )
Theorem	ismkvnn 15697*	The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- ( A e. V -> ( A e. Markov <-> A. f e. ( { 0 , 1 } ^m A ) ( -. A. x e. A ( f ` x ) = 1 -> E. x e. A ( f ` x ) = 0 ) ) )
Theorem	redcwlpolemeq1 15698*	Lemma for redcwlpo 15699. A biconditionalized version of trilpolemeq1 15684. (Contributed by Jim Kingdon, 21-Jun-2024.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   =>    |- ( ph -> ( A = 1 <-> A. x e. NN ( F ` x ) = 1 ) )
Theorem	redcwlpo 15699*	Decidability of real number equality implies the Weak Limited Principle of Omniscience (WLPO). 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 is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 15698). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones. Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO". WLPO 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 qdceq 10334 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A. x e. RR A. y e. RR DECID x = y -> om e. WOmni )
Theorem	tridceq 15700*	Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 15687 and redcwlpo 15699). Thus, this is an analytic analogue to lpowlpo 7234. (Contributed by Jim Kingdon, 24-Jul-2024.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) -> A. x e. RR A. y e. RR DECID x = y )