P. 74 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xpdjuen 7301	Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. ( B ⊔ C ) ) ~~ ( ( A X. B ) ⊔ ( A X. C ) ) )
Theorem	djudoml 7302	A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
|- ( ( A e. V /\ B e. W ) -> A ~<_ ( A ⊔ B ) )
Theorem	djudomr 7303	A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
|- ( ( A e. V /\ B e. W ) -> B ~<_ ( A ⊔ B ) )
2.6.44  Ordinal trichotomy
Theorem	exmidontriimlem1 7304	Lemma for exmidontriim 7308. A variation of r19.30dc 2644. (Contributed by Jim Kingdon, 12-Aug-2024.)
|- ( ( A. x e. A ( ph \/ ps \/ ch ) /\ EXMID ) -> ( E. x e. A ph \/ E. x e. A ps \/ A. x e. A ch ) )
Theorem	exmidontriimlem2 7305*	Lemma for exmidontriim 7308. (Contributed by Jim Kingdon, 12-Aug-2024.)
|- ( ph -> B e. On )   &    |- ( ph -> EXMID )   &    |- ( ph -> A. y e. B ( A e. y \/ A = y \/ y e. A ) )   =>    |- ( ph -> ( A e. B \/ A. y e. B y e. A ) )
Theorem	exmidontriimlem3 7306*	Lemma for exmidontriim 7308. What we get to do based on induction on both A and B. (Contributed by Jim Kingdon, 10-Aug-2024.)
|- ( ph -> A e. On )   &    |- ( ph -> B e. On )   &    |- ( ph -> EXMID )   &    |- ( ph -> A. z e. A A. y e. On ( z e. y \/ z = y \/ y e. z ) )   &    |- ( ph -> A. y e. B ( A e. y \/ A = y \/ y e. A ) )   =>    |- ( ph -> ( A e. B \/ A = B \/ B e. A ) )
Theorem	exmidontriimlem4 7307*	Lemma for exmidontriim 7308. The induction step for the induction on A. (Contributed by Jim Kingdon, 10-Aug-2024.)
|- ( ph -> A e. On )   &    |- ( ph -> B e. On )   &    |- ( ph -> EXMID )   &    |- ( ph -> A. z e. A A. y e. On ( z e. y \/ z = y \/ y e. z ) )   =>    |- ( ph -> ( A e. B \/ A = B \/ B e. A ) )
Theorem	exmidontriim 7308*	Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.)
|- (EXMID -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
2.6.45  Excluded middle and the power set of a singleton
Theorem	pw1on 7309	The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
|- ~P 1o e. On
Theorem	pw1dom2 7310	The power set of 1o dominates 2o. Also see pwpw0ss 3835 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.)
|- 2o ~<_ ~P 1o
Theorem	pw1ne0 7311	The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= (/)
Theorem	pw1ne1 7312	The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= 1o
Theorem	pw1ne3 7313	The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= 3o
Theorem	pw1nel3 7314	Negated excluded middle implies that the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- ( -. EXMID -> -. ~P 1o e. 3o )
Theorem	sucpw1ne3 7315	Negated excluded middle implies that the successor of the power set of 1o is not three . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- ( -. EXMID -> suc ~P 1o =/= 3o )
Theorem	sucpw1nel3 7316	The successor of the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- -. suc ~P 1o e. 3o
Theorem	3nelsucpw1 7317	Three is not an element of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- -. 3o e. suc ~P 1o
Theorem	sucpw1nss3 7318	Negated excluded middle implies that the successor of the power set of 1o is not a subset of 3o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
|- ( -. EXMID -> -. suc ~P 1o C_ 3o )
Theorem	3nsssucpw1 7319	Negated excluded middle implies that 3o is not a subset of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
|- ( -. EXMID -> -. 3o C_ suc ~P 1o )
Theorem	onntri35 7320*	Double negated ordinal trichotomy. There are five equivalent statements: (1) -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ), (2) -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ), (3) A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ), (4) A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ), and (5) -. -. EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7321), (3) implies (5) (onntri35 7320), (5) implies (1) (onntri51 7323), (2) implies (4) (onntri24 7325), (4) implies (5) (onntri45 7324), and (5) implies (2) (onntri52 7327). Another way of stating this is that EXMID is equivalent to trichotomy, either the x e. y \/ x = y \/ y e. x or the x C_ y \/ y C_ x form, as shown in exmidontri 7322 and exmidontri2or 7326, respectively. Thus -. -. EXMID is equivalent to (1) or (2). In addition, -. -. EXMID is equivalent to (3) by onntri3or 7328 and (4) by onntri2or 7329. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) -> -. -. EXMID )
Theorem	onntri13 7321	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) )
Theorem	exmidontri 7322*	Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- (EXMID <-> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
Theorem	onntri51 7323*	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. EXMID -> -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
Theorem	onntri45 7324*	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) -> -. -. EXMID )
Theorem	onntri24 7325	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) )
Theorem	exmidontri2or 7326*	Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
Theorem	onntri52 7327*	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. EXMID -> -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
Theorem	onntri3or 7328*	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
|- ( -. -. EXMID <-> A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) )
Theorem	onntri2or 7329*	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
|- ( -. -. EXMID <-> A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) )
2.6.46  Apartness relations
Syntax	wap 7330	Apartness predicate symbol.
wff R Ap A
Definition	df-pap 7331*	Apartness predicate. A relation R is an apartness if it is irreflexive, symmetric, and cotransitive. (Contributed by Jim Kingdon, 14-Feb-2025.)
|- ( R Ap 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 ) ) ) ) )
Syntax	wtap 7332	Tight apartness predicate symbol.
wff R TAp A
Definition	df-tap 7333*	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	dftap2 7334*	Tight apartness with the apartness properties from df-pap 7331 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 7335	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
|- ( R = S -> ( R TAp A <-> S TAp A ) )
Theorem	tapeq2 7336	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
|- ( A = B -> ( R TAp A <-> R TAp B ) )
Theorem	netap 7337*	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 7338*	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 7339*	(/) 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 7340*	Lemma for 2omotap 7342. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( -. -. ph -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o )
Theorem	2omotaplemst 7341*	Lemma for 2omotap 7342. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( ( E* r r TAp 2o /\ -. -. ph ) -> ph )
Theorem	2omotap 7342	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 7343*	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 7344*	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 7289 but since it implies excluded middle as shown at exmidac 7292, 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 7345	Formula for an abbreviation of countable choice.
wff CCHOICE
Definition	df-cc 7346*	The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7289 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 7347*	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 7348*	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 7349*	Lemma for cc2 7350. (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 7350*	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 7351*	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 7352*	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 7353*	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 7354*	Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7353, 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 7355	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 6541 and similar theorems ), going from there to positive integers (df-ni 7388) and then positive rational numbers (df-nqqs 7432) 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 7550. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 8016 and the MacNeille reals fail to satisfy axltwlin 8111, 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 7356	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 7357	Positive integer addition.
class +N
Syntax	cmi 7358	Positive integer multiplication.
class .N
Syntax	clti 7359	Positive integer ordering relation.
class <N
Syntax	cplpq 7360	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 7361	Positive pre-fraction multiplication.
class .pQ
Syntax	cltpq 7362	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 7363	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 7364	Set of positive fractions.
class Q.
Syntax	c1q 7365	The positive fraction constant 1.
class 1Q
Syntax	cplq 7366	Positive fraction addition.
class +Q
Syntax	cmq 7367	Positive fraction multiplication.
class .Q
Syntax	crq 7368	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 7369	Positive fraction ordering relation.
class <Q
Syntax	ceq0 7370	Equivalence class used to construct nonnegative fractions.
class ~Q0
Syntax	cnq0 7371	Set of nonnegative fractions.
class Q0
Syntax	c0q0 7372	The nonnegative fraction constant 0.
class 0Q0
Syntax	cplq0 7373	Nonnegative fraction addition.
class +Q0
Syntax	cmq0 7374	Nonnegative fraction multiplication.
class ·Q0
Syntax	cnp 7375	Set of positive reals.
class P.
Syntax	c1p 7376	Positive real constant 1.
class 1P
Syntax	cpp 7377	Positive real addition.
class +P.
Syntax	cmp 7378	Positive real multiplication.
class .P.
Syntax	cltp 7379	Positive real ordering relation.
class <P
Syntax	cer 7380	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 7381	Set of signed reals.
class R.
Syntax	c0r 7382	The signed real constant 0.
class 0R
Syntax	c1r 7383	The signed real constant 1.
class 1R
Syntax	cm1r 7384	The signed real constant -1.
class -1R
Syntax	cplr 7385	Signed real addition.
class +R
Syntax	cmr 7386	Signed real multiplication.
class .R
Syntax	cltr 7387	Signed real ordering relation.
class <R
Definition	df-ni 7388	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 7389	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 7390	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 7391	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 7392	Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
|- ( A e. N. <-> ( A e. om /\ A =/= (/) ) )
Theorem	pinn 7393	A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
|- ( A e. N. -> A e. om )
Theorem	pion 7394	A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
|- ( A e. N. -> A e. On )
Theorem	piord 7395	A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
|- ( A e. N. -> Ord A )
Theorem	niex 7396	The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
|- N. e. _V
Theorem	0npi 7397	The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
|- -. (/) e. N.
Theorem	elni2 7398	Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
|- ( A e. N. <-> ( A e. om /\ (/) e. A ) )
Theorem	1pi 7399	Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
|- 1o e. N.
Theorem	addpiord 7400	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 ) )