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Theorem List for Metamath Proof Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfindsg 4901* 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 hypothesis. The basis of this version is an arbitrary natural number  B instead of zero. (Contributed by NM, 16-Sep-1995.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ( B  e.  om  ->  ps )   &    |-  (
 ( ( y  e. 
 om  /\  B  e.  om )  /\  B  C_  y )  ->  ( ch 
 ->  th ) )   =>    |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ta )
 
Theoremfinds2 4902* 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 hypothesis. 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 4903* 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 hypothesis. 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 4904 Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. See tfindes 4871 for the transfinite version. (Contributed by Raph Levien, 9-Jul-2003.)
 |-  [. (/)  /  x ]. ph   &    |-  ( x  e.  om  ->  (
 ph  ->  [. suc  x  /  x ]. ph ) )   =>    |-  ( x  e.  om  ->  ph )
 
2.4.7  Relations
 
Syntaxcxp 4905 Extend the definition of a class to include the cross product.
 class  ( A  X.  B )
 
Syntaxccnv 4906 Extend the definition of a class to include the converse of a class.
 class  `' A
 
Syntaxcdm 4907 Extend the definition of a class to include the domain of a class.
 class  dom  A
 
Syntaxcrn 4908 Extend the definition of a class to include the range of a class.
 class  ran  A
 
Syntaxcres 4909 Extend the definition of a class to include the restriction of a class. (Read: The restriction of  A to  B.)
 class  ( A  |`  B )
 
Syntaxcima 4910 Extend the definition of a class to include the image of a class. (Read: The image of  B under  A.)
 class  ( A " B )
 
Syntaxccom 4911 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 4912 Extend the definition of a wff to include the relation predicate. (Read:  A is a relation.)
 wff  Rel  A
 
Definitiondf-xp 4913* 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
>. } ) (ex-xp 21775). Another example is that the set of rational numbers are defined in df-q 10606 using the cross-product  ( ZZ  X.  NN ); 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 4914 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5350 and dfrel3 5357. (Contributed by NM, 1-Aug-1994.)
 |-  ( Rel  A  <->  A  C_  ( _V 
 X.  _V ) )
 
Definitiondf-cnv 4915* 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 5084 (see df-br 4238 and df-rel 4914 for more on relations). For example,  `' { <. 2 ,  6 >. , 
<. 3 ,  9
>. }  =  { <. 6 ,  2 >. , 
<. 9 ,  3
>. } (ex-cnv 21776). 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 4916* Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example,  ( ( exp 
o.  cos ) `  0
)  =  _e (ex-co 21777) because  ( cos `  0 )  =  1 (see cos0 12782) and  ( exp `  1
)  =  _e (see df-e 12702). Note that Definition 7 of [Suppes] p. 63 reverses  A and  B, uses  /. 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 4917* Define the domain of a class. Definition 3 of [Suppes] p. 59. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  dom  F  =  { 2 ,  3 } (ex-dm 21778). Another example is the domain of the complex arctangent,  ( A  e. 
dom arctan 
<->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/=  _i ) ) (for proof see atandm 20747). Contrast with range (defined in df-rn 4918). For alternate definitions see dfdm2 5430, dfdm3 5087, and dfdm4 5092. 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 4918 Define the range of a class. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  ran  F  =  { 6 ,  9 } (ex-rn 21779). Contrast with domain (defined in df-dm 4917). For alternate definitions, see dfrn2 5088, dfrn3 5089, and dfrn4 5360. 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 4919 Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, the expression  ( exp  |`  RR ) (used in reeff1 12752) means "the exponential function e to the x, but the exponent x must be in the reals" (df-ef 12701 defines the exponential function, which normally allows the exponent to be a complex number). Another example is that  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  /\  B  =  { 1 ,  2 } )  ->  ( F  |`  B )  =  { <. 2 ,  6
>. } (ex-res 21780). (Contributed by NM, 2-Aug-1994.)
 |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V )
 )
 
Definitiondf-ima 4920 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 } (ex-ima 21781). Contrast with restriction (df-res 4919) and range (df-rn 4918). For an alternate definition, see dfima2 5234. (Contributed by NM, 2-Aug-1994.)
 |-  ( A " B )  =  ran  ( A  |`  B )
 
Theoremxpeq1 4921 Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2 4922 Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
 |-  ( A  =  B  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremelxpi 4923* Membership in a cross product. Uses fewer axioms than elxp 4924. (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 4924* 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 4925* 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 4926 Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C )  =  ( B  X.  D ) )
 
Theoremxpeq1i 4927 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A  X.  C )  =  ( B  X.  C )
 
Theoremxpeq2i 4928 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C  X.  A )  =  ( C  X.  B )
 
Theoremxpeq12i 4929 Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  X.  C )  =  ( B  X.  D )
 
Theoremxpeq1d 4930 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2d 4931 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
Theoremxpeq12d 4932 Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  X.  C )  =  ( B  X.  D ) )
 
Theoremnfxp 4933 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 )
 
Theoremcsbxpg 4934 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 ) )
 
Theorem0nelxp 4935 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 4936 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 4937 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 4938 Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)
 |-  ( A ( C  X.  D ) B  <-> 
 ( A  e.  C  /\  B  e.  D ) )
 
Theoremopelxpi 4939 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 ) )
 
Theoremopelxp1 4940 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 4941 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 4942 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 4943* 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 4944 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 )
 )
 
Theorembrrelex 4945 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 4946 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 )
 
Theorembrrelexi 4947 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 4948 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 4949 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
 
Theoremfconstmpt 4950* 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 4951* 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 4952 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 4953 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 4954 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 4955 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 4956* 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 4957* 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 4958 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 4959 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 4960 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 4961* 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 4962* 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 4963* 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 4964 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 4965* 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 4966* 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 4967 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
 
Theorembrinxp2 4968 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 4969 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 4970 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 4971 Intersection of total 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 )
 
Theoremfrinxp 4972 Intersection of well-founded relation with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
 |-  ( R  Fr  A  <->  ( R  i^i  ( A  X.  A ) )  Fr  A )
 
Theoremseinxp 4973 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 )
 
Theoremweinxp 4974 Intersection of well-ordering with cross product of its field. (Contributed by NM, 9-Mar-1997.) (Revised by Mario Carneiro, 10-Jul-2014.)
 |-  ( R  We  A  <->  ( R  i^i  ( A  X.  A ) )  We  A )
 
Theoremposn 4975 Partial ordering of a singleton. (Contributed by NM, 27-Apr-2009.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( Rel  R  ->  ( R  Po  { A } 
 <->  -.  A R A ) )
 
Theoremsosn 4976 Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( Rel  R  ->  ( R  Or  { A } 
 <->  -.  A R A ) )
 
Theoremfrsn 4977 Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( Rel  R  ->  ( R  Fr  { A } 
 <->  -.  A R A ) )
 
Theoremwesn 4978 Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( Rel  R  ->  ( R  We  { A } 
 <->  -.  A R A ) )
 
Theoremopabssxp 4979* 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 4980* The law of concretion for a binary relation. See brab2a 4956 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 4981* 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 4982* 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 4983* 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 4984* 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 ) )
 
Theoremxp0r 4985 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 )  =  (/)
 
Theoremonxpdisj 4986 Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 4729. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( On  i^i  ( _V  X.  _V ) )  =  (/)
 
Theoremonnev 4987 The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.)
 |- 
 On  =/=  _V
 
Theoremreleq 4988 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremreleqi 4989 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
 |-  A  =  B   =>    |-  ( Rel  A  <->  Rel 
 B )
 
Theoremreleqd 4990 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Rel  A  <->  Rel  B ) )
 
Theoremnfrel 4991 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
 
Theoremrelss 4992 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 4993* 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 4994* 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 4995* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4993 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 4996* 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 4997* 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 4998* 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 )
 
Theoremeqrelriiv 4999* Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)
 |- 
 Rel  A   &    |-  Rel  B   &    |-  ( <. x ,  y >.  e.  A  <->  <. x ,  y >.  e.  B )   =>    |-  A  =  B
 
Theoremeqbrriv 5000* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
 |- 
 Rel  A   &    |-  Rel  B   &    |-  ( x A y  <->  x B y )   =>    |-  A  =  B
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