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Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0eln0 4501 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 4502* If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4418 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6361 or nntri3or 6355), 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 4503* 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 4504 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 4447) but also holds when it is  (/) by uni0 3731. (Contributed by Jim Kingdon, 31-Jul-2022.)
 |-  ( A  e.  om  ->  U. A  e.  om )
 
2.6.6  Relations
 
Syntaxcxp 4505 Extend the definition of a class to include the cross product.
 class  ( A  X.  B )
 
Syntaxccnv 4506 Extend the definition of a class to include the converse of a class.
 class  `' A
 
Syntaxcdm 4507 Extend the definition of a class to include the domain of a class.
 class  dom  A
 
Syntaxcrn 4508 Extend the definition of a class to include the range of a class.
 class  ran  A
 
Syntaxcres 4509 Extend the definition of a class to include the restriction of a class. (Read: The restriction of  A to  B.)
 class  ( A  |`  B )
 
Syntaxcima 4510 Extend the definition of a class to include the image of a class. (Read: The image of  B under  A.)
 class  ( A " B )
 
Syntaxccom 4511 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 4512 Extend the definition of a wff to include the relation predicate. (Read:  A is a relation.)
 wff  Rel  A
 
Definitiondf-xp 4513* 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 4514 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4957 and dfrel3 4964. (Contributed by NM, 1-Aug-1994.)
 |-  ( Rel  A  <->  A  C_  ( _V 
 X.  _V ) )
 
Definitiondf-cnv 4515* 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 4690 (see df-br 3898 and df-rel 4514 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 4516* 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 4517* 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 4518). For alternate definitions see dfdm2 5041, dfdm3 4694, and dfdm4 4699. 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 4518 Define the range of a class. For example, F = {  <. 2 , 6  >.,  <. 3 , 9  >. } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4517). For alternate definitions, see dfrn2 4695, dfrn3 4696, and dfrn4 4967. 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 4519 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 4520 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 4519) and range (df-rn 4518). For an alternate definition, see dfima2 4851. (Contributed by NM, 2-Aug-1994.)
 |-  ( A " B )  =  ran  ( A  |`  B )
 
Theoremxpeq1 4521 Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2 4522 Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
 |-  ( A  =  B  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremelxpi 4523* Membership in a cross product. Uses fewer axioms than elxp 4524. (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 4524* 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 4525* 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 4526 Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C )  =  ( B  X.  D ) )
 
Theoremxpeq1i 4527 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A  X.  C )  =  ( B  X.  C )
 
Theoremxpeq2i 4528 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C  X.  A )  =  ( C  X.  B )
 
Theoremxpeq12i 4529 Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  X.  C )  =  ( B  X.  D )
 
Theoremxpeq1d 4530 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2d 4531 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremxpeq12d 4532 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 4533 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 4534 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 4535 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 4536 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 4537 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 4538 Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)
 |-  ( A ( C  X.  D ) B  <-> 
 ( A  e.  C  /\  B  e.  D ) )
 
Theoremopelxpi 4539 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 4540 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 4541 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 4542 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 4543 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 4544* 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 4545 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 4546 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 4547 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 4548 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 4549 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 4550 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 4551 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 4552 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 4553 A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
 |-  ( Rel  R  ->  (/)  e/  R )
 
Theoremfconstmpt 4554* 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 4555* 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 4556 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 4557 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 4558 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 4559 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 4560* 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 4561* 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 4562 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 4563 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 4564 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 4565* 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 4566* 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 4567* 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 4568 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 4569* 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 4570* 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 4571 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
 
Theoremmosubopt 4572* "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 4573* "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 4574 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 4575 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 4576 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 4577 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 4578 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 4579 Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
 |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Po  { A }  <->  -.  A R A ) )
 
Theoremsosng 4580 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 4581* 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 4582* The law of concretion for a binary relation. See brab2a 4560 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 4583* 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 4584* 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 4585* 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 4586* 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 4587 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 4588 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 4589 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremreleqi 4590 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
 |-  A  =  B   =>    |-  ( Rel  A  <->  Rel 
 B )
 
Theoremreleqd 4591 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremnfrel 4592 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 4593 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 4594 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 4595* 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 4596* 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 4597* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4595 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 4598* 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
 
Theoremrelssdv 4599* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  ( <. x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremeqrelriv 4600* Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
 |-  ( <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B )   =>    |-  ( ( Rel 
 A  /\  Rel  B ) 
 ->  A  =  B )
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