P. 77 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	papirr 7601	An apartness is irreflexive. (Contributed by Jim Kingdon, 27-May-2026.)
|- ( ( R Ap A /\ X e. A ) -> -. X R X )
Theorem	papsym 7602	An apartness is symmetric. (Contributed by Jim Kingdon, 27-May-2026.)
|- ( ph -> R Ap A )   &    |- ( ph -> X e. A )   &    |- ( ph -> Y e. A )   &    |- ( ph -> X R Y )   =>    |- ( ph -> Y R X )
Theorem	papcotr 7603	An apartness is cotransitive. (Contributed by Jim Kingdon, 28-May-2026.)
|- ( ph -> R Ap A )   &    |- ( ph -> X e. A )   &    |- ( ph -> Y e. A )   &    |- ( ph -> X R Y )   &    |- ( ph -> Z e. A )   =>    |- ( ph -> ( X R Z \/ Y R Z ) )
Syntax	wtap 7604	Tight apartness predicate symbol.
wff R TAp A
Definition	df-tap 7605*	Tight apartness predicate. A relation R is a tight apartness if it is irreflexive, symmetric, cotransitive, and tight. (Contributed by Jim Kingdon, 5-Feb-2025.)
|- ( R TAp A <-> ( R Ap A /\ A. x e. A A. y e. A ( -. x R y -> x = y ) ) )
Theorem	tapap 7606	A tight apartness is an apartness. (Contributed by Jim Kingdon, 29-May-2026.)
|- ( R TAp A -> R Ap A )
Theorem	dftap2 7607*	Tight apartness with the apartness properties from df-pap 7598 expanded. (Contributed by Jim Kingdon, 21-Feb-2025.)
|- ( R TAp A <-> ( R C_ ( A X. A ) /\ ( A. x e. A -. x R x /\ A. x e. A A. y e. A ( x R y -> y R x ) ) /\ ( A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) /\ A. x e. A A. y e. A ( -. x R y -> x = y ) ) ) )
Theorem	tapeq1 7608	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
|- ( R = S -> ( R TAp A <-> S TAp A ) )
Theorem	tapeq2 7609	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
|- ( A = B -> ( R TAp A <-> R TAp B ) )
Theorem	netap 7610*	Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.)
|- ( A. x e. A A. y e. A DECID x = y -> { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } TAp A )
Theorem	2onetap 7611*	Negated equality is a tight apartness on 2o. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } TAp 2o
Theorem	2oneel 7612*	(/) and 1o are two unequal elements of 2o. (Contributed by Jim Kingdon, 8-Feb-2025.)
|- <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) }
Theorem	2omotaplemap 7613*	Lemma for 2omotap 7615. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( -. -. ph -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o )
Theorem	2omotaplemst 7614*	Lemma for 2omotap 7615. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( ( E* r r TAp 2o /\ -. -. ph ) -> ph )
Theorem	2omotap 7615	If there is at most one tight apartness on 2o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( E* r r TAp 2o -> EXMID )
Theorem	exmidapne 7616*	Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.)
|- (EXMID -> ( R TAp A <-> R = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) )
Theorem	exmidmotap 7617*	The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.)
|- (EXMID <-> A. x E* r r TAp x )
PART 3  CHOICE PRINCIPLES We have already introduced the full Axiom of Choice df-ac 7552 but since it implies excluded middle as shown at exmidac 7555, it is not especially relevant to us. In this section we define countable choice and dependent choice, which are not as strong as thus often considered in mathematics which seeks to avoid full excluded middle.
3.1  Countable Choice and Dependent Choice
3.1.1  Introduce Countable Choice
Syntax	wacc 7618	Formula for an abbreviation of countable choice.
wff CCHOICE
Definition	df-cc 7619*	The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7552 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
|- (CCHOICE <-> A. x ( dom x ~~ om -> E. f ( f C_ x /\ f Fn dom x ) ) )
Theorem	ccfunen 7620*	Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A ~~ om )   &    |- ( ph -> A. x e. A E. w w e. x )   =>    |- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )
Theorem	cc1 7621*	Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
|- (CCHOICE -> A. x ( ( x ~~ om /\ A. z e. x E. w w e. z ) -> E. f A. z e. x ( f ` z ) e. z ) )
Theorem	cc2lem 7622*	Lemma for cc2 7623. (Contributed by Jim Kingdon, 27-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> F Fn om )   &    |- ( ph -> A. x e. om E. w w e. ( F ` x ) )   &    |- A = ( n e. om |-> ( { n } X. ( F ` n ) ) )   &    |- G = ( n e. om |-> ( 2nd ` ( f ` ( A ` n ) ) ) )   =>    |- ( ph -> E. g ( g Fn om /\ A. n e. om ( g ` n ) e. ( F ` n ) ) )
Theorem	cc2 7623*	Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> F Fn om )   &    |- ( ph -> A. x e. om E. w w e. ( F ` x ) )   =>    |- ( ph -> E. g ( g Fn om /\ A. n e. om ( g ` n ) e. ( F ` n ) ) )
Theorem	cc3 7624*	Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A. n e. N F e. _V )   &    |- ( ph -> A. n e. N E. w w e. F )   &    |- ( ph -> N ~~ om )   =>    |- ( ph -> E. f ( f Fn N /\ A. n e. N ( f ` n ) e. F ) )
Theorem	cc4f 7625*	Countable choice by showing the existence of a function f which can choose a value at each index n such that ch holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A e. V )   &    |- F/_ n A   &    |- ( ph -> N ~~ om )   &    |- ( x = ( f ` n ) -> ( ps <-> ch ) )   &    |- ( ph -> A. n e. N E. x e. A ps )   =>    |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )
Theorem	cc4 7626*	Countable choice by showing the existence of a function f which can choose a value at each index n such that ch holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 1-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A e. V )   &    |- ( ph -> N ~~ om )   &    |- ( x = ( f ` n ) -> ( ps <-> ch ) )   &    |- ( ph -> A. n e. N E. x e. A ps )   =>    |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )
Theorem	cc4n 7627*	Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7626, the hypotheses only require an A(n) for each value of n, not a single set A which suffices for every n e. om. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A. n e. N { x e. A | ps } e. V )   &    |- ( ph -> N ~~ om )   &    |- ( x = ( f ` n ) -> ( ps <-> ch ) )   &    |- ( ph -> A. n e. N E. x e. A ps )   =>    |- ( ph -> E. f ( f Fn N /\ A. n e. N ch ) )
Theorem	acnccim 7628	Given countable choice, every set has choice sets of length om. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- (CCHOICE -> AC om = _V )
PART 4  REAL AND COMPLEX NUMBERS This section derives the basics of real and complex numbers. To construct the real numbers constructively, we follow two main sources. The first is Metamath Proof Explorer, which has the advantage of being already formalized in metamath. Its disadvantage, for our purposes, is that it assumes the law of the excluded middle throughout. Since we have already developed natural numbers ( for example, nna0 6737 and similar theorems ), going from there to positive integers (df-ni 7661) and then positive rational numbers (df-nqqs 7705) does not involve a major change in approach compared with the Metamath Proof Explorer. It is when we proceed to Dedekind cuts that we bring in more material from Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which are different without excluded middle or choice principles. With excluded middle, it is natural to define a cut as the lower set only (as Metamath Proof Explorer does), but here we define the cut as a pair of both the lower and upper sets, as [HoTT] does. There are also differences in how we handle order and replacing "not equal to zero" with "apart from zero". When working constructively, there are several possible definitions of real numbers. Here we adopt the most common definition, as two-sided Dedekind cuts with the properties described at df-inp 7823. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 8289 and the MacNeille reals fail to satisfy axltwlin 8383, and we do not develop them here. For more on differing definitions of the reals, see the introduction to Chapter 11 in [HoTT] or Section 1.2 of [BauerHanson].
4.1  Construction and axiomatization of real and complex numbers
4.1.1  Dedekind-cut construction of real and complex numbers
Syntax	cnpi 7629	The set of positive integers, which is the set of natural numbers om with 0 removed. Note: This is the start of the Dedekind-cut construction of real and complex numbers.
class N.
Syntax	cpli 7630	Positive integer addition.
class +N
Syntax	cmi 7631	Positive integer multiplication.
class .N
Syntax	clti 7632	Positive integer ordering relation.
class <N
Syntax	cplpq 7633	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 7634	Positive pre-fraction multiplication.
class .pQ
Syntax	cltpq 7635	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 7636	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 7637	Set of positive fractions.
class Q.
Syntax	c1q 7638	The positive fraction constant 1.
class 1Q
Syntax	cplq 7639	Positive fraction addition.
class +Q
Syntax	cmq 7640	Positive fraction multiplication.
class .Q
Syntax	crq 7641	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 7642	Positive fraction ordering relation.
class <Q
Syntax	ceq0 7643	Equivalence class used to construct nonnegative fractions.
class ~Q0
Syntax	cnq0 7644	Set of nonnegative fractions.
class Q0
Syntax	c0q0 7645	The nonnegative fraction constant 0.
class 0Q0
Syntax	cplq0 7646	Nonnegative fraction addition.
class +Q0
Syntax	cmq0 7647	Nonnegative fraction multiplication.
class ·Q0
Syntax	cnp 7648	Set of positive reals.
class P.
Syntax	c1p 7649	Positive real constant 1.
class 1P
Syntax	cpp 7650	Positive real addition.
class +P.
Syntax	cmp 7651	Positive real multiplication.
class .P.
Syntax	cltp 7652	Positive real ordering relation.
class <P
Syntax	cer 7653	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 7654	Set of signed reals.
class R.
Syntax	c0r 7655	The signed real constant 0.
class 0R
Syntax	c1r 7656	The signed real constant 1.
class 1R
Syntax	cm1r 7657	The signed real constant -1.
class -1R
Syntax	cplr 7658	Signed real addition.
class +R
Syntax	cmr 7659	Signed real multiplication.
class .R
Syntax	cltr 7660	Signed real ordering relation.
class <R
Definition	df-ni 7661	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.)
|- N. = ( om \ { (/) } )
Definition	df-pli 7662	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)
|- +N = ( +o |` ( N. X. N. ) )
Definition	df-mi 7663	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)
|- .N = ( .o |` ( N. X. N. ) )
Definition	df-lti 7664	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.)
|- <N = ( _E i^i ( N. X. N. ) )
Theorem	elni 7665	Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
|- ( A e. N. <-> ( A e. om /\ A =/= (/) ) )
Theorem	pinn 7666	A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
|- ( A e. N. -> A e. om )
Theorem	pion 7667	A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
|- ( A e. N. -> A e. On )
Theorem	piord 7668	A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
|- ( A e. N. -> Ord A )
Theorem	niex 7669	The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
|- N. e. _V
Theorem	0npi 7670	The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
|- -. (/) e. N.
Theorem	elni2 7671	Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
|- ( A e. N. <-> ( A e. om /\ (/) e. A ) )
Theorem	1pi 7672	Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
|- 1o e. N.
Theorem	addpiord 7673	Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.)
|- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( A +o B ) )
Theorem	mulpiord 7674	Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
|- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) = ( A .o B ) )
Theorem	mulidpi 7675	1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ( A e. N. -> ( A .N 1o ) = A )
Theorem	ltpiord 7676	Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> A e. B ) )
Theorem	ltsopi 7677	Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.)
|- <N Or N.
Theorem	pitric 7678	Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> -. ( A = B \/ B <N A ) ) )
Theorem	pitri3or 7679	Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B \/ A = B \/ B <N A ) )
Theorem	ltdcpi 7680	Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
|- ( ( A e. N. /\ B e. N. ) -> DECID A <N B )
Theorem	ltrelpi 7681	Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
|- <N C_ ( N. X. N. )
Theorem	dmaddpi 7682	Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
|- dom +N = ( N. X. N. )
Theorem	dmmulpi 7683	Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
|- dom .N = ( N. X. N. )
Theorem	addclpi 7684	Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
|- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) e. N. )
Theorem	mulclpi 7685	Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
|- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) e. N. )
Theorem	addcompig 7686	Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( B +N A ) )
Theorem	addasspig 7687	Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A +N B ) +N C ) = ( A +N ( B +N C ) ) )
Theorem	mulcompig 7688	Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) = ( B .N A ) )
Theorem	mulasspig 7689	Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A .N B ) .N C ) = ( A .N ( B .N C ) ) )
Theorem	distrpig 7690	Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( A .N ( B +N C ) ) = ( ( A .N B ) +N ( A .N C ) ) )
Theorem	addcanpig 7691	Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A +N B ) = ( A +N C ) <-> B = C ) )
Theorem	mulcanpig 7692	Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A .N B ) = ( A .N C ) <-> B = C ) )
Theorem	addnidpig 7693	There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.)
|- ( ( A e. N. /\ B e. N. ) -> -. ( A +N B ) = A )
Theorem	ltexpi 7694*	Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> E. x e. N. ( A +N x ) = B ) )
Theorem	ltapig 7695	Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( A <N B <-> ( C +N A ) <N ( C +N B ) ) )
Theorem	ltmpig 7696	Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( A <N B <-> ( C .N A ) <N ( C .N B ) ) )
Theorem	1lt2pi 7697	One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
|- 1o <N ( 1o +N 1o )
Theorem	nlt1pig 7698	No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( A e. N. -> -. A <N 1o )
Theorem	indpi 7699*	Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.)
|- ( x = 1o -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y +N 1o ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. N. -> ( ch -> th ) )   =>    |- ( A e. N. -> ta )
Theorem	nnppipi 7700	A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.)
|- ( ( A e. om /\ B e. N. ) -> ( A +o B ) e. N. )