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Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfinds2 4601* 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 )
 )
 
Theoremfinds1 4602* 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 )
 
Theoremfindes 4603 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)
 
Theoremnn0suc 4604* 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 ) )
 
Theoremelomssom 4605 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 4606. (Revised by BJ, 7-Aug-2024.)
 |-  ( A  e.  om  ->  A  C_  om )
 
Theoremelnn 4606 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 )
 
Theoremordom 4607 Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
 |- 
 Ord  om
 
Theoremomelon2 4608 Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
 |-  ( om  e.  _V  ->  om  e.  On )
 
Theoremomelon 4609 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
 |- 
 om  e.  On
 
Theoremnnon 4610 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
 |-  ( A  e.  om  ->  A  e.  On )
 
Theoremnnoni 4611 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
 |-  A  e.  om   =>    |-  A  e.  On
 
Theoremnnord 4612 A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
 |-  ( A  e.  om  ->  Ord  A )
 
Theoremomsson 4613 Omega is a subset of  On. (Contributed by NM, 13-Jun-1994.)
 |- 
 om  C_  On
 
Theoremlimom 4614 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
 
Theorempeano2b 4615 A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
 |-  ( A  e.  om  <->  suc  A  e.  om )
 
Theoremnnsuc 4616* A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
 |-  ( ( A  e.  om 
 /\  A  =/=  (/) )  ->  E. x  e.  om  A  =  suc  x )
 
Theoremnnsucpred 4617 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 )
 
Theoremnndceq0 4618 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  =  (/) )
 
Theorem0elnn 4619 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 ) )
 
Theoremnn0eln0 4620 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  =/=  (/) ) )
 
Theoremnnregexmid 4621* If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4535 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6500 or nntri3or 6494), 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 )
 
Theoremomsinds 4622* 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 )
 
Theoremnnpredcl 4623 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 4564) but also holds when it is  (/) by uni0 3837. (Contributed by Jim Kingdon, 31-Jul-2022.)
 |-  ( A  e.  om  ->  U. A  e.  om )
 
Theoremnnpredlt 4624 The predecessor (see nnpredcl 4623) of a nonzero natural number is less than (see df-iord 4367) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
 |-  ( ( A  e.  om 
 /\  A  =/=  (/) )  ->  U. A  e.  A )
 
2.6.6  Relations
 
Syntaxcxp 4625 Extend the definition of a class to include the cross product.
 class  ( A  X.  B )
 
Syntaxccnv 4626 Extend the definition of a class to include the converse of a class.
 class  `' A
 
Syntaxcdm 4627 Extend the definition of a class to include the domain of a class.
 class  dom  A
 
Syntaxcrn 4628 Extend the definition of a class to include the range of a class.
 class  ran  A
 
Syntaxcres 4629 Extend the definition of a class to include the restriction of a class. (Read: The restriction of  A to  B.)
 class  ( A  |`  B )
 
Syntaxcima 4630 Extend the definition of a class to include the image of a class. (Read: The image of  B under  A.)
 class  ( A " B )
 
Syntaxccom 4631 Extend the definition of a class to include the composition of two classes. (Read: The composition of  A and  B.)
 class  ( A  o.  B )
 
Syntaxwrel 4632 Extend the definition of a wff to include the relation predicate. (Read:  A is a relation.)
 wff  Rel  A
 
Definitiondf-xp 4633* 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 ) }
 
Definitiondf-rel 4634 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5080 and dfrel3 5087. (Contributed by NM, 1-Aug-1994.)
 |-  ( Rel  A  <->  A  C_  ( _V 
 X.  _V ) )
 
Definitiondf-cnv 4635* 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 4811 (see df-br 4005 and df-rel 4634 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 }
 
Definitiondf-co 4636* 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 ) }
 
Definitiondf-dm 4637* 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 4638). For alternate definitions see dfdm2 5164, dfdm3 4815, and dfdm4 4820. 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 }
 
Definitiondf-rn 4638 Define the range of a class. For example, F = {  <. 2 , 6  >.,  <. 3 , 9  >. } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4637). For alternate definitions, see dfrn2 4816, dfrn3 4817, and dfrn4 5090. 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
 
Definitiondf-res 4639 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 )
 )
 
Definitiondf-ima 4640 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 4639) and range (df-rn 4638). For an alternate definition, see dfima2 4973. (Contributed by NM, 2-Aug-1994.)
 |-  ( A " B )  =  ran  ( A  |`  B )
 
Theoremxpeq1 4641 Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2 4642 Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
 |-  ( A  =  B  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremelxpi 4643* Membership in a cross product. Uses fewer axioms than elxp 4644. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  e.  ( B  X.  C )  ->  E. x E. y ( A  =  <. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C ) ) )
 
Theoremelxp 4644* 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 )
 ) )
 
Theoremelxp2 4645* 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 >. )
 
Theoremxpeq12 4646 Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C )  =  ( B  X.  D ) )
 
Theoremxpeq1i 4647 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A  X.  C )  =  ( B  X.  C )
 
Theoremxpeq2i 4648 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C  X.  A )  =  ( C  X.  B )
 
Theoremxpeq12i 4649 Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  X.  C )  =  ( B  X.  D )
 
Theoremxpeq1d 4650 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2d 4651 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremxpeq12d 4652 Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  X.  C )  =  ( B  X.  D ) )
 
Theoremsqxpeqd 4653 Equality deduction for a Cartesian square, see Wikipedia "Cartesian product", https://en.wikipedia.org/wiki/Cartesian_product#n-ary_Cartesian_power. (Contributed by AV, 13-Jan-2020.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  X.  A )  =  ( B  X.  B ) )
 
Theoremnfxp 4654 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 )
 
Theorem0nelxp 4655 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 )
 
Theorem0nelelxp 4656 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 )
 
Theoremopelxp 4657 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 ) )
 
Theorembrxp 4658 Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)
 |-  ( A ( C  X.  D ) B  <-> 
 ( A  e.  C  /\  B  e.  D ) )
 
Theoremopelxpi 4659 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 ) )
 
Theoremopelxpd 4660 Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
 |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  D )   =>    |-  ( ph  ->  <. A ,  B >.  e.  ( C  X.  D ) )
 
Theoremopelxp1 4661 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 )
 
Theoremopelxp2 4662 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 )
 
Theoremotelxp1 4663 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 )
 
Theoremrabxp 4664* 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 ) }
 
Theorembrrelex12 4665 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 )
 )
 
Theorembrrelex1 4666 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 )
 
Theorembrrelex 4667 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 )
 
Theorembrrelex2 4668 A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( Rel  R  /\  A R B ) 
 ->  B  e.  _V )
 
Theorembrrelex12i 4669 Two classes that are related by a binary relation are sets. (An artifact of our ordered pair definition.) (Contributed by BJ, 3-Oct-2022.)
 |- 
 Rel  R   =>    |-  ( A R B  ->  ( A  e.  _V  /\  B  e.  _V )
 )
 
Theorembrrelex1i 4670 The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)
 |- 
 Rel  R   =>    |-  ( A R B  ->  A  e.  _V )
 
Theorembrrelex2i 4671 The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 Rel  R   =>    |-  ( A R B  ->  B  e.  _V )
 
Theoremnprrel 4672 No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
 |- 
 Rel  R   &    |-  -.  A  e.  _V   =>    |-  -.  A R B
 
Theorem0nelrel 4673 A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
 |-  ( Rel  R  ->  (/)  e/  R )
 
Theoremfconstmpt 4674* Representation of a constant function using the mapping operation. (Note that  x cannot appear free in  B.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |-  ( A  X.  { B } )  =  ( x  e.  A  |->  B )
 
Theoremvtoclr 4675* Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |- 
 Rel  R   &    |-  ( ( x R y  /\  y R z )  ->  x R z )   =>    |-  ( ( A R B  /\  B R C )  ->  A R C )
 
Theoremopelvvg 4676 Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  <. A ,  B >.  e.  ( _V  X.  _V ) )
 
Theoremopelvv 4677 Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 <. A ,  B >.  e.  ( _V  X.  _V )
 
Theoremopthprc 4678 Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.)
 |-  ( ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } )
 )  =  ( ( C  X.  { (/) } )  u.  ( D  X.  { { (/) } } )
 ) 
 <->  ( A  =  C  /\  B  =  D ) )
 
Theorembrel 4679 Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  R  C_  ( C  X.  D )   =>    |-  ( A R B  ->  ( A  e.  C  /\  B  e.  D ) )
 
Theorembrab2a 4680* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)
 |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D )  /\  ph ) }   =>    |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D ) 
 /\  ps ) )
 
Theoremelxp3 4681* Membership in a cross product. (Contributed by NM, 5-Mar-1995.)
 |-  ( A  e.  ( B  X.  C )  <->  E. x E. y
 ( <. x ,  y >.  =  A  /\  <. x ,  y >.  e.  ( B  X.  C ) ) )
 
Theoremopeliunxp 4682 Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
 |-  ( <. x ,  C >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  C  e.  B ) )
 
Theoremxpundi 4683 Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
 |-  ( A  X.  ( B  u.  C ) )  =  ( ( A  X.  B )  u.  ( A  X.  C ) )
 
Theoremxpundir 4684 Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
 |-  ( ( A  u.  B )  X.  C )  =  ( ( A  X.  C )  u.  ( B  X.  C ) )
 
Theoremxpiundi 4685* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( C  X.  U_ x  e.  A  B )  =  U_ x  e.  A  ( C  X.  B )
 
Theoremxpiundir 4686* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( U_ x  e.  A  B  X.  C )  =  U_ x  e.  A  ( B  X.  C )
 
Theoremiunxpconst 4687* Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  U_ x  e.  A  ( { x }  X.  B )  =  ( A  X.  B )
 
Theoremxpun 4688 The cross product of two unions. (Contributed by NM, 12-Aug-2004.)
 |-  ( ( A  u.  B )  X.  ( C  u.  D ) )  =  ( ( ( A  X.  C )  u.  ( A  X.  D ) )  u.  ( ( B  X.  C )  u.  ( B  X.  D ) ) )
 
Theoremelvv 4689* Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
 
Theoremelvvv 4690* Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
 |-  ( A  e.  (
 ( _V  X.  _V )  X.  _V )  <->  E. x E. y E. z  A  =  <.
 <. x ,  y >. ,  z >. )
 
Theoremelvvuni 4691 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
 
Theoremmosubopt 4692* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
 |-  ( A. y A. z E* x ph  ->  E* x E. y E. z ( A  =  <. y ,  z >.  /\  ph ) )
 
Theoremmosubop 4693* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
 |- 
 E* x ph   =>    |- 
 E* x E. y E. z ( A  =  <. y ,  z >.  /\  ph )
 
Theorembrinxp2 4694 Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A ( R  i^i  ( C  X.  D ) ) B  <-> 
 ( A  e.  C  /\  B  e.  D  /\  A R B ) )
 
Theorembrinxp 4695 Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B 
 <->  A ( R  i^i  ( C  X.  D ) ) B ) )
 
Theorempoinxp 4696 Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
 |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A ) )  Po  A )
 
Theoremsoinxp 4697 Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
 |-  ( R  Or  A  <->  ( R  i^i  ( A  X.  A ) )  Or  A )
 
Theoremseinxp 4698 Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( R Se  A  <->  ( R  i^i  ( A  X.  A ) ) Se  A )
 
Theoremposng 4699 Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
 |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Po  { A }  <->  -.  A R A ) )
 
Theoremsosng 4700 Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
 |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Or  { A }  <->  -.  A R A ) )
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