P. 44 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4301-4400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	suc11 4301	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 4302*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3962 can also be summarized as "at least two sets exist", the difference is that dtruarb 3962 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 4303*	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 4302. (Contributed by Jim Kingdon, 29-Dec-2018.)
|- -. A. x x = y
Theorem	eunex 4304	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 4305	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 4306	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 4307	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 4308*	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 4309	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 4310	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 4126 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 4187) and U. { x e. On | x C_ A } = A (onuniss2 4256). Constructively ( ~P A i^i On ) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4313). (Contributed by Jim Kingdon, 21-Jul-2019.)
|- ( Ord A -> suc A C_ ( ~P A i^i On ) )
Theorem	onnmin 4311	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 4312	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 4313*	The subset in ordpwsucss 4310 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 4314*	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	onintexmid 4315*	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 4316	The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
|- _E Fr A
Theorem	ordfr 4317	Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
|- ( Ord A -> _E Fr A )
Theorem	ordwe 4318	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 4319*	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 4320	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 4321*	Lemma for reg3exmid 4322. Our counterexample A satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- _E We A
Theorem	reg3exmid 4322*	If any inhabited set satisfying df-wetr 4089 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 )
2.5.3  Transfinite induction
Theorem	tfi 4323*	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 4324*	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 4325*	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 4326*	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 4327*	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 4328*	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 4329*	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 4330*	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 (i.e. finite ordinals)
Syntax	com 4331	Extend class notation to include the class of natural numbers.
class om
Definition	df-iom 4332*	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 4123. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers with analogous properties and operations, but they will be different sets. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4333 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 4333*	Alias for df-iom 4332. Use it instead of df-iom 4332 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 4334	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 4335	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 4336	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 4337	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 4338	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 4339*	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 4344. (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 4340*	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 4341*	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 4342*	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 4343*	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 4344	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 4345*	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	elnn 4346	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 4347	Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
|- Ord om
Theorem	omelon2 4348	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
|- ( om e. _V -> om e. On )
Theorem	omelon 4349	Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
|- om e. On
Theorem	nnon 4350	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
|- ( A e. om -> A e. On )
Theorem	nnoni 4351	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
|- A e. om   =>    |- A e. On
Theorem	nnord 4352	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
|- ( A e. om -> Ord A )
Theorem	omsson 4353	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.)
|- om C_ On
Theorem	limom 4354	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 4355	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
|- ( A e. om <-> suc A e. om )
Theorem	nnsuc 4356*	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	nndceq0 4357	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 4358	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 4359	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 4360*	If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4278 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6100 or nntri3or 6095), 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 )
2.6.6  Relations
Syntax	cxp 4361	Extend the definition of a class to include the cross product.
class ( A X. B )
Syntax	ccnv 4362	Extend the definition of a class to include the converse of a class.
class `' A
Syntax	cdm 4363	Extend the definition of a class to include the domain of a class.
class dom A
Syntax	crn 4364	Extend the definition of a class to include the range of a class.
class ran A
Syntax	cres 4365	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 4366	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 4367	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 4368	Extend the definition of a wff to include the relation predicate. (Read: A is a relation.)
wff Rel A
Definition	df-xp 4369*	Define the cross product of two classes. 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 are defined in using the cross-product ( Z X. N ) ; the left- and right-hand sides of the cross-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 4370	Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4791 and dfrel3 4798. (Contributed by NM, 1-Aug-1994.)
|- ( Rel A <-> A C_ ( _V X. _V ) )
Definition	df-cnv 4371*	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 4536 (see df-br 3786 and df-rel 4370 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. Many authors use the postfix superscript "to the minus one." "Converse" is Quine's terminology; some authors call it "inverse," especially when the argument is a function. (Contributed by NM, 4-Jul-1994.)
|- `' A = { <. x , y >. | y A x }
Definition	df-co 4372*	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 4373*	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 4374). For alternate definitions see dfdm2 4872, dfdm3 4540, and dfdm4 4545. 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 4374	Define the range of a class. For example, F = { <. 2 , 6 >., <. 3 , 9 >. } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4373). For alternate definitions, see dfrn2 4541, dfrn3 4542, and dfrn4 4801. 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 4375	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 >. } . (Contributed by NM, 2-Aug-1994.)
|- ( A |` B ) = ( A i^i ( B X. _V ) )
Definition	df-ima 4376	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 4375) and range (df-rn 4374). For an alternate definition, see dfima2 4690. (Contributed by NM, 2-Aug-1994.)
|- ( A " B ) = ran ( A |` B )
Theorem	xpeq1 4377	Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
|- ( A = B -> ( A X. C ) = ( B X. C ) )
Theorem	xpeq2 4378	Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
|- ( A = B -> ( C X. A ) = ( C X. B ) )
Theorem	elxpi 4379*	Membership in a cross product. Uses fewer axioms than elxp 4380. (Contributed by NM, 4-Jul-1994.)
|- ( A e. ( B X. C ) -> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )
Theorem	elxp 4380*	Membership in a cross product. (Contributed by NM, 4-Jul-1994.)
|- ( A e. ( B X. C ) <-> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )
Theorem	elxp2 4381*	Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
|- ( A e. ( B X. C ) <-> E. x e. B E. y e. C A = <. x , y >. )
Theorem	xpeq12 4382	Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
|- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )
Theorem	xpeq1i 4383	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( A X. C ) = ( B X. C )
Theorem	xpeq2i 4384	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( C X. A ) = ( C X. B )
Theorem	xpeq12i 4385	Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
|- A = B   &    |- C = D   =>    |- ( A X. C ) = ( B X. D )
Theorem	xpeq1d 4386	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A X. C ) = ( B X. C ) )
Theorem	xpeq2d 4387	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C X. A ) = ( C X. B ) )
Theorem	xpeq12d 4388	Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A X. C ) = ( B X. D ) )
Theorem	nfxp 4389	Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( A X. B )
Theorem	0nelxp 4390	The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- -. (/) e. ( A X. B )
Theorem	0nelelxp 4391	A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)
|- ( C e. ( A X. B ) -> -. (/) e. C )
Theorem	opelxp 4392	Ordered pair membership in a cross product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( <. A , B >. e. ( C X. D ) <-> ( A e. C /\ B e. D ) )
Theorem	brxp 4393	Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)
|- ( A ( C X. D ) B <-> ( A e. C /\ B e. D ) )
Theorem	opelxpi 4394	Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.)
|- ( ( A e. C /\ B e. D ) -> <. A , B >. e. ( C X. D ) )
Theorem	opelxp1 4395	The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( <. A , B >. e. ( C X. D ) -> A e. C )
Theorem	opelxp2 4396	The second member of an ordered pair of classes in a cross product belongs to second cross product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( <. A , B >. e. ( C X. D ) -> B e. D )
Theorem	otelxp1 4397	The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.)
|- ( <. <. A , B >. , C >. e. ( ( R X. S ) X. T ) -> A e. R )
Theorem	rabxp 4398*	Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)
|- ( x = <. y , z >. -> ( ph <-> ps ) )   =>    |- { x e. ( A X. B ) | ph } = { <. y , z >. | ( y e. A /\ z e. B /\ ps ) }
Theorem	brrelex12 4399	A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )
Theorem	brrelex 4400	A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( Rel R /\ A R B ) -> A e. _V )