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Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnndceq0 4501 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 4502 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 4503 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 4504* If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4420 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6363 or nntri3or 6357), 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 4505* 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 4506 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 4449) but also holds when it is  (/) by uni0 3733. (Contributed by Jim Kingdon, 31-Jul-2022.)
 |-  ( A  e.  om  ->  U. A  e.  om )
 
2.6.6  Relations
 
Syntaxcxp 4507 Extend the definition of a class to include the cross product.
 class  ( A  X.  B )
 
Syntaxccnv 4508 Extend the definition of a class to include the converse of a class.
 class  `' A
 
Syntaxcdm 4509 Extend the definition of a class to include the domain of a class.
 class  dom  A
 
Syntaxcrn 4510 Extend the definition of a class to include the range of a class.
 class  ran  A
 
Syntaxcres 4511 Extend the definition of a class to include the restriction of a class. (Read: The restriction of  A to  B.)
 class  ( A  |`  B )
 
Syntaxcima 4512 Extend the definition of a class to include the image of a class. (Read: The image of  B under  A.)
 class  ( A " B )
 
Syntaxccom 4513 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 4514 Extend the definition of a wff to include the relation predicate. (Read:  A is a relation.)
 wff  Rel  A
 
Definitiondf-xp 4515* 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 ) }
 
Definitiondf-rel 4516 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4959 and dfrel3 4966. (Contributed by NM, 1-Aug-1994.)
 |-  ( Rel  A  <->  A  C_  ( _V 
 X.  _V ) )
 
Definitiondf-cnv 4517* 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 4692 (see df-br 3900 and df-rel 4516 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 }
 
Definitiondf-co 4518* 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 4519* 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 4520). For alternate definitions see dfdm2 5043, dfdm3 4696, and dfdm4 4701. 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 4520 Define the range of a class. For example, F = {  <. 2 , 6  >.,  <. 3 , 9  >. } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4519). For alternate definitions, see dfrn2 4697, dfrn3 4698, and dfrn4 4969. 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 4521 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 )
 )
 
Definitiondf-ima 4522 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 4521) and range (df-rn 4520). For an alternate definition, see dfima2 4853. (Contributed by NM, 2-Aug-1994.)
 |-  ( A " B )  =  ran  ( A  |`  B )
 
Theoremxpeq1 4523 Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2 4524 Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
 |-  ( A  =  B  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremelxpi 4525* Membership in a cross product. Uses fewer axioms than elxp 4526. (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 4526* 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 4527* 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 4528 Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C )  =  ( B  X.  D ) )
 
Theoremxpeq1i 4529 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A  X.  C )  =  ( B  X.  C )
 
Theoremxpeq2i 4530 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C  X.  A )  =  ( C  X.  B )
 
Theoremxpeq12i 4531 Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  X.  C )  =  ( B  X.  D )
 
Theoremxpeq1d 4532 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2d 4533 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremxpeq12d 4534 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 4535 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 4536 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 4537 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 4538 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 4539 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 4540 Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)
 |-  ( A ( C  X.  D ) B  <-> 
 ( A  e.  C  /\  B  e.  D ) )
 
Theoremopelxpi 4541 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 4542 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 4543 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 4544 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 4545 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 4546* 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 4547 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 4548 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 4549 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 4550 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 4551 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 4552 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 4553 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 4554 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 4555 A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
 |-  ( Rel  R  ->  (/)  e/  R )
 
Theoremfconstmpt 4556* 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 4557* 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 4558 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 4559 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 4560 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 4561 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 4562* 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 4563* 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 4564 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 4565 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 4566 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 4567* 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 4568* 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 4569* 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 4570 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 4571* 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 4572* 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 4573 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
 
Theoremmosubopt 4574* "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 4575* "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 4576 Intersection of binary relation with cross 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 4577 Intersection of binary relation with cross 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 4578 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 4579 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 4580 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 4581 Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
 |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Po  { A }  <->  -.  A R A ) )
 
Theoremsosng 4582 Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
 |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Or  { A }  <->  -.  A R A ) )
 
Theoremopabssxp 4583* An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)
 |- 
 { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  C_  ( A  X.  B )
 
Theorembrab2ga 4584* The law of concretion for a binary relation. See brab2a 4562 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)
 |-  ( ( 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 ) )
 
Theoremoptocl 4585* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
 |-  D  =  ( B  X.  C )   &    |-  ( <. x ,  y >.  =  A  ->  ( ph  <->  ps ) )   &    |-  ( ( x  e.  B  /\  y  e.  C )  ->  ph )   =>    |-  ( A  e.  D  ->  ps )
 
Theorem2optocl 4586* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
 |-  R  =  ( C  X.  D )   &    |-  ( <. x ,  y >.  =  A  ->  ( ph  <->  ps ) )   &    |-  ( <. z ,  w >.  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 ( ( x  e.  C  /\  y  e.  D )  /\  (
 z  e.  C  /\  w  e.  D )
 )  ->  ph )   =>    |-  ( ( A  e.  R  /\  B  e.  R )  ->  ch )
 
Theorem3optocl 4587* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
 |-  R  =  ( D  X.  F )   &    |-  ( <. x ,  y >.  =  A  ->  ( ph  <->  ps ) )   &    |-  ( <. z ,  w >.  =  B  ->  ( ps  <->  ch ) )   &    |-  ( <. v ,  u >.  =  C  ->  ( ch  <->  th ) )   &    |-  ( ( ( x  e.  D  /\  y  e.  F )  /\  ( z  e.  D  /\  w  e.  F )  /\  ( v  e.  D  /\  u  e.  F ) )  ->  ph )   =>    |-  ( ( A  e.  R  /\  B  e.  R  /\  C  e.  R ) 
 ->  th )
 
Theoremopbrop 4588* Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)
 |-  ( ( ( z  =  A  /\  w  =  B )  /\  (
 v  =  C  /\  u  =  D )
 )  ->  ( ph  <->  ps ) )   &    |-  R  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) ) 
 /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ph )
 ) }   =>    |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  ( <. A ,  B >. R <. C ,  D >.  <->  ps ) )
 
Theorem0xp 4589 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)
 |-  ( (/)  X.  A )  =  (/)
 
Theoremcsbxpg 4590 Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  D  -> 
 [_ A  /  x ]_ ( B  X.  C )  =  ( [_ A  /  x ]_ B  X.  [_ A  /  x ]_ C ) )
 
Theoremreleq 4591 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremreleqi 4592 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
 |-  A  =  B   =>    |-  ( Rel  A  <->  Rel 
 B )
 
Theoremreleqd 4593 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremnfrel 4594 Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x Rel  A
 
Theoremsbcrel 4595 Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. Rel  R  <->  Rel  [_ A  /  x ]_ R ) )
 
Theoremrelss 4596 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)
 |-  ( A  C_  B  ->  ( Rel  B  ->  Rel 
 A ) )
 
Theoremssrel 4597* A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( Rel  A  ->  ( A  C_  B  <->  A. x A. y
 ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B ) ) )
 
Theoremeqrel 4598* Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)
 |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y (
 <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B ) ) )
 
Theoremssrel2 4599* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4597 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)
 |-  ( R  C_  ( A  X.  B )  ->  ( R  C_  S  <->  A. x  e.  A  A. y  e.  B  (
 <. x ,  y >.  e.  R  ->  <. x ,  y >.  e.  S ) ) )
 
Theoremrelssi 4600* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
 |- 
 Rel  A   &    |-  ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B )   =>    |-  A  C_  B
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