P. 46 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elirr 4501	No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. The reason that this theorem is marked as discouraged is a bit subtle. If we wanted to reduce usage of ax-setind 4497, we could redefine Ord A (df-iord 4327) to also require _E Fr A (df-frind 4293) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4502 (which under that definition would presumably not need ax-setind 4497 to prove it). But since ordinals have not yet been defined that way, we cannot rely on the "don't add additional axiom use" feature of the minimizer to get theorems to use ordirr 4502. To encourage ordirr 4502 when possible, we mark this theorem as discouraged. (Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.)
|- -. A e. A
Theorem	ordirr 4502	Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The present proof requires ax-setind 4497. If in the definition of ordinals df-iord 4327, we also required that membership be well-founded on any ordinal (see df-frind 4293), then we could prove ordirr 4502 without ax-setind 4497. (Contributed by NM, 2-Jan-1994.)
|- ( Ord A -> -. A e. A )
Theorem	onirri 4503	An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
|- A e. On   =>    |- -. A e. A
Theorem	nordeq 4504	A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
|- ( ( Ord A /\ B e. A ) -> A =/= B )
Theorem	ordn2lp 4505	An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)
|- ( Ord A -> -. ( A e. B /\ B e. A ) )
Theorem	orddisj 4506	An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
|- ( Ord A -> ( A i^i { A } ) = (/) )
Theorem	orddif 4507	Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
|- ( Ord A -> A = ( suc A \ { A } ) )
Theorem	elirrv 4508	The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.)
|- -. x e. x
Theorem	sucprcreg 4509	A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.)
|- ( -. A e. _V <-> suc A = A )
Theorem	ruv 4510	The Russell class is equal to the universe _V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
|- { x | x e/ x } = _V
Theorem	ruALT 4511	Alternate proof of Russell's Paradox ru 2936, simplified using (indirectly) the Axiom of Set Induction ax-setind 4497. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
|- { x | x e/ x } e/ _V
Theorem	onprc 4512	No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4446), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.)
|- -. On e. _V
Theorem	sucon 4513	The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
|- suc On = On
Theorem	en2lp 4514	No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 27-Nov-2018.)
|- -. ( A e. B /\ B e. A )
Theorem	preleq 4515	Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) -> ( A = C /\ B = D ) )
Theorem	opthreg 4516	Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4497 (via the preleq 4515 step). See df-op 3569 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( { A , { A , B } } = { C , { C , D } } <-> ( A = C /\ B = D ) )
Theorem	suc11g 4517	The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
|- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) )
Theorem	suc11 4518	The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
|- ( ( A e. On /\ B e. On ) -> ( suc A = suc B <-> A = B ) )
Theorem	dtruex 4519*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4153 can also be summarized as "at least two sets exist", the difference is that dtruarb 4153 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific y, we can construct a set x which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- E. x -. x = y
Theorem	dtru 4520*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4519. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- -. A. x x = y
Theorem	eunex 4521	Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- ( E! x ph -> E. x -. ph )
Theorem	ordsoexmid 4522	Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.)
|- _E Or On   =>    |- ( ph \/ -. ph )
Theorem	ordsuc 4523	The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.)
|- ( Ord A <-> Ord suc A )
Theorem	onsucuni2 4524	A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( A e. On /\ A = suc B ) -> suc U. A = A )
Theorem	0elsucexmid 4525*	If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
|- A. x e. On (/) e. suc x   =>    |- ( ph \/ -. ph )
Theorem	nlimsucg 4526	A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A e. V -> -. Lim suc A )
Theorem	ordpwsucss 4527	The collection of ordinals in the power class of an ordinal is a superset of its successor. We can think of ( ~P A i^i On ) as another possible definition of successor, which would be equivalent to df-suc 4332 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A e. On then both U. suc A = A (onunisuci 4393) and U. { x e. On | x C_ A } = A (onuniss2 4472). Constructively ( ~P A i^i On ) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4530). (Contributed by Jim Kingdon, 21-Jul-2019.)
|- ( Ord A -> suc A C_ ( ~P A i^i On ) )
Theorem	onnmin 4528	No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.)
|- ( ( A C_ On /\ B e. A ) -> -. B e. |^| A )
Theorem	ssnel 4529	Relationship between subset and elementhood. In the context of ordinals this can be seen as an ordering law. (Contributed by Jim Kingdon, 22-Jul-2019.)
|- ( A C_ B -> -. B e. A )
Theorem	ordpwsucexmid 4530*	The subset in ordpwsucss 4527 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.)
|- A. x e. On suc x = ( ~P x i^i On )   =>    |- ( ph \/ -. ph )
Theorem	ordtri2or2exmid 4531*	Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- A. x e. On A. y e. On ( x C_ y \/ y C_ x )   =>    |- ( ph \/ -. ph )
Theorem	ontri2orexmidim 4532*	Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4531. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> DECID ph )
Theorem	onintexmid 4533*	If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
|- ( ( y C_ On /\ E. x x e. y ) -> |^| y e. y )   =>    |- ( ph \/ -. ph )
Theorem	zfregfr 4534	The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
|- _E Fr A
Theorem	ordfr 4535	Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
|- ( Ord A -> _E Fr A )
Theorem	ordwe 4536	Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
|- ( Ord A -> _E We A )
Theorem	wetriext 4537*	A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
|- ( ph -> R We A )   &    |- ( ph -> A e. V )   &    |- ( ph -> A. a e. A A. b e. A ( a R b \/ a = b \/ b R a ) )   &    |- ( ph -> B e. A )   &    |- ( ph -> C e. A )   &    |- ( ph -> A. z e. A ( z R B <-> z R C ) )   =>    |- ( ph -> B = C )
Theorem	wessep 4538	A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)
|- ( ( _E We A /\ B C_ A ) -> _E We B )
Theorem	reg3exmidlemwe 4539*	Lemma for reg3exmid 4540. Our counterexample A satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- _E We A
Theorem	reg3exmid 4540*	If any inhabited set satisfying df-wetr 4295 for _E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- ( ( _E We z /\ E. w w e. z ) -> E. x e. z A. y e. z x C_ y )   =>    |- ( ph \/ -. ph )
Theorem	dcextest 4541*	If it is decidable whether { x | ph } is a set, then -. ph is decidable (where x does not occur in ph). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition -. ph is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)
|- DECID { x | ph } e. _V   =>    |- DECID -. ph
2.5.3  Transfinite induction
Theorem	tfi 4542*	The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if A is a class of ordinal numbers with the property that every ordinal number included in A also belongs to A, then every ordinal number is in A. (Contributed by NM, 18-Feb-2004.)
|- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On )
Theorem	tfis 4543*	Transfinite Induction Schema. If all ordinal numbers less than a given number x have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.)
|- ( x e. On -> ( A. y e. x [ y / x ] ph -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis2f 4544*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis2 4545*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis3 4546*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = A -> ( ph <-> ch ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( A e. On -> ch )
Theorem	tfisi 4547*	A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.)
|- ( ph -> A e. V )   &    |- ( ph -> T e. On )   &    |- ( ( ph /\ ( R e. On /\ R C_ T ) /\ A. y ( S e. R -> ch ) ) -> ps )   &    |- ( x = y -> ( ps <-> ch ) )   &    |- ( x = A -> ( ps <-> th ) )   &    |- ( x = y -> R = S )   &    |- ( x = A -> R = T )   =>    |- ( ph -> th )
2.6  IZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity
Axiom	ax-iinf 4548*	Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.)
|- E. x ( (/) e. x /\ A. y ( y e. x -> suc y e. x ) )
Theorem	zfinf2 4549*	A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.)
|- E. x ( (/) e. x /\ A. y e. x suc y e. x )
2.6.2  The natural numbers
Syntax	com 4550	Extend class notation to include the class of natural numbers.
class om
Definition	df-iom 4551*	Define the class of natural numbers as the smallest inductive set, which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. Note: the natural numbers om are a subset of the ordinal numbers df-on 4329. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8835) with analogous properties and operations, but they will be different sets. We are unable to use the terms finite ordinal and natural number interchangeably, as shown at exmidonfin 7130. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4552 instead for naming consistency with set.mm. (New usage is discouraged.)
|- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
Theorem	dfom3 4552*	Alias for df-iom 4551. Use it instead of df-iom 4551 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.)
|- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
Theorem	omex 4553	The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.)
|- om e. _V
2.6.3  Peano's postulates
Theorem	peano1 4554	Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.)
|- (/) e. om
Theorem	peano2 4555	The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
|- ( A e. om -> suc A e. om )
Theorem	peano3 4556	The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
|- ( A e. om -> suc A =/= (/) )
Theorem	peano4 4557	Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.)
|- ( ( A e. om /\ B e. om ) -> ( suc A = suc B <-> A = B ) )
Theorem	peano5 4558*	The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as Theorem findes 4563. (Contributed by NM, 18-Feb-2004.)
|- ( ( (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A )
2.6.4  Finite induction (for finite ordinals)
Theorem	find 4559*	The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that A is a set of natural numbers, zero belongs to A, and given any member of A the member's successor also belongs to A. The conclusion is that every natural number is in A. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A )   =>    |- A = om
Theorem	finds 4560*	Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = suc y -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. om -> ( ch -> th ) )   =>    |- ( A e. om -> ta )
Theorem	finds2 4561*	Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = suc y -> ( ph <-> th ) )   &    |- ( ta -> ps )   &    |- ( y e. om -> ( ta -> ( ch -> th ) ) )   =>    |- ( x e. om -> ( ta -> ph ) )
Theorem	finds1 4562*	Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = suc y -> ( ph <-> th ) )   &    |- ps   &    |- ( y e. om -> ( ch -> th ) )   =>    |- ( x e. om -> ph )
Theorem	findes 4563	Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)
|- [. (/) / x ]. ph   &    |- ( x e. om -> ( ph -> [. suc x / x ]. ph ) )   =>    |- ( x e. om -> ph )
2.6.5  The Natural Numbers (continued)
Theorem	nn0suc 4564*	A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.)
|- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )
Theorem	elomssom 4565	A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4566. (Revised by BJ, 7-Aug-2024.)
|- ( A e. om -> A C_ om )
Theorem	elnn 4566	A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
|- ( ( A e. B /\ B e. om ) -> A e. om )
Theorem	ordom 4567	Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
|- Ord om
Theorem	omelon2 4568	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
|- ( om e. _V -> om e. On )
Theorem	omelon 4569	Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
|- om e. On
Theorem	nnon 4570	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
|- ( A e. om -> A e. On )
Theorem	nnoni 4571	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
|- A e. om   =>    |- A e. On
Theorem	nnord 4572	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
|- ( A e. om -> Ord A )
Theorem	omsson 4573	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.)
|- om C_ On
Theorem	limom 4574	Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. (Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon, 5-Jan-2019.)
|- Lim om
Theorem	peano2b 4575	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
|- ( A e. om <-> suc A e. om )
Theorem	nnsuc 4576*	A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
|- ( ( A e. om /\ A =/= (/) ) -> E. x e. om A = suc x )
Theorem	nnsucpred 4577	The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- ( ( A e. om /\ A =/= (/) ) -> suc U. A = A )
Theorem	nndceq0 4578	A natural number is either zero or nonzero. Decidable equality for natural numbers is a special case of the law of the excluded middle which holds in most constructive set theories including ours. (Contributed by Jim Kingdon, 5-Jan-2019.)
|- ( A e. om -> DECID A = (/) )
Theorem	0elnn 4579	A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.)
|- ( A e. om -> ( A = (/) \/ (/) e. A ) )
Theorem	nn0eln0 4580	A natural number is nonempty iff it contains the empty set. Although in constructive mathematics it is generally more natural to work with inhabited sets and ignore the whole concept of nonempty sets, in the specific case of natural numbers this theorem may be helpful in converting proofs which were written assuming excluded middle. (Contributed by Jim Kingdon, 28-Aug-2019.)
|- ( A e. om -> ( (/) e. A <-> A =/= (/) ) )
Theorem	nnregexmid 4581*	If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4495 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6447 or nntri3or 6441), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.)
|- ( ( x C_ om /\ E. y y e. x ) -> E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) )   =>    |- ( ph \/ -. ph )
Theorem	omsinds 4582*	Strong (or "total") induction principle over om. (Contributed by Scott Fenton, 17-Jul-2015.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = A -> ( ph <-> ch ) )   &    |- ( x e. om -> ( A. y e. x ps -> ph ) )   =>    |- ( A e. om -> ch )
Theorem	nnpredcl 4583	The predecessor of a natural number is a natural number. This theorem is most interesting when the natural number is a successor (as seen in theorems like onsucuni2 4524) but also holds when it is (/) by uni0 3800. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- ( A e. om -> U. A e. om )
Theorem	nnpredlt 4584	The predecessor (see nnpredcl 4583) of a nonzero natural number is less than (see df-iord 4327) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
|- ( ( A e. om /\ A =/= (/) ) -> U. A e. A )
2.6.6  Relations
Syntax	cxp 4585	Extend the definition of a class to include the cross product.
class ( A X. B )
Syntax	ccnv 4586	Extend the definition of a class to include the converse of a class.
class `' A
Syntax	cdm 4587	Extend the definition of a class to include the domain of a class.
class dom A
Syntax	crn 4588	Extend the definition of a class to include the range of a class.
class ran A
Syntax	cres 4589	Extend the definition of a class to include the restriction of a class. (Read: The restriction of A to B.)
class ( A |` B )
Syntax	cima 4590	Extend the definition of a class to include the image of a class. (Read: The image of B under A.)
class ( A " B )
Syntax	ccom 4591	Extend the definition of a class to include the composition of two classes. (Read: The composition of A and B.)
class ( A o. B )
Syntax	wrel 4592	Extend the definition of a wff to include the relation predicate. (Read: A is a relation.)
wff Rel A
Definition	df-xp 4593*	Define the Cartesian product of two classes. This is also sometimes called the "cross product" but that term also has other meanings; we intentionally choose a less ambiguous term. Definition 9.11 of [Quine] p. 64. For example, ( { 1 , 5 } X. { 2 , 7 } ) = ( { <. 1 , 2 >. , <. 1 , 7 >. } u. { <. 5 , 2 >. , <. 5 , 7 >. } ). Another example is that the set of rational numbers is defined using the Cartesian product as ( ZZ X. NN ); the left- and right-hand sides of the Cartesian product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)
|- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
Definition	df-rel 4594	Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5037 and dfrel3 5044. (Contributed by NM, 1-Aug-1994.)
|- ( Rel A <-> A C_ ( _V X. _V ) )
Definition	df-cnv 4595*	Define the converse of a class. Definition 9.12 of [Quine] p. 64. The converse of a binary relation swaps its arguments, i.e., if A e. _V and B e. _V then ( A `' R B <-> B R A ), as proven in brcnv 4770 (see df-br 3967 and df-rel 4594 for more on relations). For example, `' { <. 2 , 6 >. , <. 3 , 9 >. } = { <. 6 , 2 >. , <. 9 , 3 >. }. We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. "Converse" is Quine's terminology. Some authors use a "minus one" exponent and call it "inverse", especially when the argument is a function, although this is not in general a genuine inverse. (Contributed by NM, 4-Jul-1994.)
|- `' A = { <. x , y >. | y A x }
Definition	df-co 4596*	Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. Note that Definition 7 of [Suppes] p. 63 reverses A and B, uses a slash instead of o., and calls the operation "relative product". (Contributed by NM, 4-Jul-1994.)
|- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) }
Definition	df-dm 4597*	Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, F = { <. 2 , 6 >., <. 3 , 9 >. } -> dom F = { 2 , 3 } . Contrast with range (defined in df-rn 4598). For alternate definitions see dfdm2 5121, dfdm3 4774, and dfdm4 4779. The notation " dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)
|- dom A = { x | E. y x A y }
Definition	df-rn 4598	Define the range of a class. For example, F = { <. 2 , 6 >., <. 3 , 9 >. } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4597). For alternate definitions, see dfrn2 4775, dfrn3 4776, and dfrn4 5047. The notation " ran " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.)
|- ran A = dom `' A
Definition	df-res 4599	Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, ( F = { <. 2 , 6 >. , <. 3 , 9 >. } /\ B = { 1 , 2 } ) -> ( F |` B ) = { <. 2 , 6 >. }. We do not introduce a special syntax for the corestriction of a class: it will be expressed either as the intersection ( A i^i ( _V X. B ) ) or as the converse of the restricted converse. (Contributed by NM, 2-Aug-1994.)
|- ( A |` B ) = ( A i^i ( B X. _V ) )
Definition	df-ima 4600	Define the image of a class (as restricted by another class). Definition 6.6(2) of [TakeutiZaring] p. 24. For example, ( F = { <. 2 , 6 >., <. 3 , 9 >. } /\ B = { 1 , 2 } ) -> ( F " B ) = { 6 } . Contrast with restriction (df-res 4599) and range (df-rn 4598). For an alternate definition, see dfima2 4931. (Contributed by NM, 2-Aug-1994.)
|- ( A " B ) = ran ( A |` B )