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Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfunrnfi 7001 The range of a finite relation is finite if its converse is a function. (Contributed by Jim Kingdon, 5-Feb-2022.)
 |-  ( ( Rel  A  /\  Fun  `' A  /\  A  e.  Fin )  ->  ran  A  e.  Fin )
 
Theoremf1ofi 7002 If a 1-1 and onto function has a finite domain, its range is finite. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  F : A -1-1-onto-> B )  ->  B  e.  Fin )
 
Theoremf1dmvrnfibi 7003 A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 7004. (Contributed by AV, 10-Jan-2020.)
 |-  ( ( A  e.  V  /\  F : A -1-1-> B )  ->  ( F  e.  Fin  <->  ran  F  e.  Fin ) )
 
Theoremf1vrnfibi 7004 A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 7003. (Contributed by AV, 10-Jan-2020.)
 |-  ( ( F  e.  V  /\  F : A -1-1-> B )  ->  ( F  e.  Fin  <->  ran  F  e.  Fin ) )
 
Theoremiunfidisj 7005* The finite union of disjoint finite sets is finite. Note that  B depends on  x, i.e. can be thought of as  B ( x ). (Contributed by NM, 23-Mar-2006.) (Revised by Jim Kingdon, 7-Oct-2022.)
 |-  ( ( A  e.  Fin  /\  A. x  e.  A  B  e.  Fin  /\ Disj  x  e.  A  B )  ->  U_ x  e.  A  B  e.  Fin )
 
Theoremf1finf1o 7006 Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.)
 |-  ( ( A  ~~  B  /\  B  e.  Fin )  ->  ( F : A -1-1-> B  <->  F : A -1-1-onto-> B ) )
 
Theoremen1eqsn 7007 A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
 |-  ( ( A  e.  B  /\  B  ~~  1o )  ->  B  =  { A } )
 
Theoremen1eqsnbi 7008 A set containing an element has exactly one element iff it is a singleton. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
 |-  ( A  e.  B  ->  ( B  ~~  1o  <->  B  =  { A } )
 )
 
Theoremsnexxph 7009* A case where the antecedent of snexg 4213 is not needed. The class  { x  | 
ph } is from dcextest 4613. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
 |- 
 { { x  |  ph
 } }  e.  _V
 
Theorempreimaf1ofi 7010 The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.)
 |-  ( ph  ->  C  C_  B )   &    |-  ( ph  ->  F : A -1-1-onto-> B )   &    |-  ( ph  ->  C  e.  Fin )   =>    |-  ( ph  ->  ( `' F " C )  e.  Fin )
 
Theoremfidcenumlemim 7011* Lemma for fidcenum 7015. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.)
 |-  ( A  e.  Fin  ->  ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  E. n  e.  om  E. f  f : n -onto-> A ) )
 
Theoremfidcenumlemrks 7012* Lemma for fidcenum 7015. Induction step for fidcenumlemrk 7013. (Contributed by Jim Kingdon, 20-Oct-2022.)
 |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )   &    |-  ( ph  ->  F : N -onto-> A )   &    |-  ( ph  ->  J  e.  om )   &    |-  ( ph  ->  suc  J  C_  N )   &    |-  ( ph  ->  ( X  e.  ( F " J )  \/  -.  X  e.  ( F " J ) ) )   &    |-  ( ph  ->  X  e.  A )   =>    |-  ( ph  ->  ( X  e.  ( F " suc  J )  \/ 
 -.  X  e.  ( F " suc  J ) ) )
 
Theoremfidcenumlemrk 7013* Lemma for fidcenum 7015. (Contributed by Jim Kingdon, 20-Oct-2022.)
 |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )   &    |-  ( ph  ->  F : N -onto-> A )   &    |-  ( ph  ->  K  e.  om )   &    |-  ( ph  ->  K  C_  N )   &    |-  ( ph  ->  X  e.  A )   =>    |-  ( ph  ->  ( X  e.  ( F " K )  \/  -.  X  e.  ( F " K ) ) )
 
Theoremfidcenumlemr 7014* Lemma for fidcenum 7015. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.)
 |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )   &    |-  ( ph  ->  F : N -onto-> A )   &    |-  ( ph  ->  N  e.  om )   =>    |-  ( ph  ->  A  e.  Fin )
 
Theoremfidcenum 7015* A set is finite if and only if it has decidable equality and is finitely enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The definition of "finitely enumerable" as  E. n  e. 
om E. f f : n -onto-> A is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.)
 |-  ( A  e.  Fin  <->  ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  E. n  e.  om  E. f  f : n -onto-> A ) )
 
2.6.32  Schroeder-Bernstein Theorem
 
Theoremsbthlem1 7016* Lemma for isbth 7026. (Contributed by NM, 22-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   =>    |-  U. D  C_  ( A  \  (
 g " ( B  \  ( f " U. D ) ) ) )
 
Theoremsbthlem2 7017* Lemma for isbth 7026. (Contributed by NM, 22-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   =>    |-  ( ran  g  C_  A  ->  ( A  \  ( g
 " ( B  \  ( f " U. D ) ) ) )  C_  U. D )
 
Theoremsbthlemi3 7018* Lemma for isbth 7026. (Contributed by NM, 22-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   =>    |-  (
 (EXMID  /\  ran  g  C_  A )  ->  ( g "
 ( B  \  (
 f " U. D ) ) )  =  ( A  \  U. D ) )
 
Theoremsbthlemi4 7019* Lemma for isbth 7026. (Contributed by NM, 27-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   =>    |-  (
 (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A 
 \  U. D ) )  =  ( B  \  ( f " U. D ) ) )
 
Theoremsbthlemi5 7020* Lemma for isbth 7026. (Contributed by NM, 22-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   =>    |-  ( (EXMID 
 /\  ( dom  f  =  A  /\  ran  g  C_  A ) )  ->  dom  H  =  A )
 
Theoremsbthlemi6 7021* Lemma for isbth 7026. (Contributed by NM, 27-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   =>    |-  ( ( (EXMID  /\  ran  f  C_  B )  /\  ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  H  =  B )
 
Theoremsbthlem7 7022* Lemma for isbth 7026. (Contributed by NM, 27-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   =>    |-  ( ( Fun  f  /\  Fun  `' g ) 
 ->  Fun  H )
 
Theoremsbthlemi8 7023* Lemma for isbth 7026. (Contributed by NM, 27-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   =>    |-  ( ( (EXMID  /\  Fun  `' f )  /\  (
 ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g
 ) )  ->  Fun  `' H )
 
Theoremsbthlemi9 7024* Lemma for isbth 7026. (Contributed by NM, 28-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   =>    |-  ( (EXMID 
 /\  f : A -1-1-> B 
 /\  g : B -1-1-> A )  ->  H : A
 -1-1-onto-> B )
 
Theoremsbthlemi10 7025* Lemma for isbth 7026. (Contributed by NM, 28-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   &    |-  B  e.  _V   =>    |-  (
 (EXMID  /\  ( A  ~<_  B  /\  B 
 ~<_  A ) )  ->  A  ~~  B )
 
Theoremisbth 7026 Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set 
A is smaller (has lower cardinality) than  B and vice-versa, then  A and  B are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 7016 through sbthlemi10 7025; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 7025. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. The proof does require the law of the excluded middle which cannot be avoided as shown at exmidsbthr 15513. (Contributed by NM, 8-Jun-1998.)
 |-  ( (EXMID 
 /\  ( A  ~<_  B  /\  B 
 ~<_  A ) )  ->  A  ~~  B )
 
2.6.33  Finite intersections
 
Syntaxcfi 7027 Extend class notation with the function whose value is the class of finite intersections of the elements of a given set.
 class  fi
 
Definitiondf-fi 7028* Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 7031). (Contributed by FL, 27-Apr-2008.)
 |- 
 fi  =  ( x  e.  _V  |->  { z  |  E. y  e.  ( ~P x  i^i  Fin )
 z  =  |^| y } )
 
Theoremfival 7029* The set of all the finite intersections of the elements of  A. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( A  e.  V  ->  ( fi `  A )  =  { y  |  E. x  e.  ( ~P A  i^i  Fin )
 y  =  |^| x } )
 
Theoremelfi 7030* Specific properties of an element of 
( fi `  B
). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
 
Theoremelfi2 7031* The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( B  e.  V  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  (
 ( ~P B  i^i  Fin )  \  { (/) } ) A  =  |^| x ) )
 
Theoremelfir 7032 Sufficient condition for an element of  ( fi `  B ). (Contributed by Mario Carneiro, 24-Nov-2013.)
 |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e.  Fin ) )  ->  |^| A  e.  ( fi
 `  B ) )
 
Theoremssfii 7033 Any element of a set  A is the intersection of a finite subset of  A. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  ( A  e.  V  ->  A  C_  ( fi `  A ) )
 
Theoremfi0 7034 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( fi `  (/) )  =  (/)
 
Theoremfieq0 7035 A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( A  e.  V  ->  ( A  =  (/)  <->  ( fi `  A )  =  (/) ) )
 
Theoremfiss 7036 Subset relationship for function 
fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( ( B  e.  V  /\  A  C_  B )  ->  ( fi `  A )  C_  ( fi
 `  B ) )
 
Theoremfiuni 7037 The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( A  e.  V  ->  U. A  =  U. ( fi `  A ) )
 
Theoremfipwssg 7038 If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( A  e.  V  /\  A  C_  ~P X )  ->  ( fi `  A )  C_  ~P X )
 
Theoremfifo 7039* Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  F  =  ( y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  |->  |^| y )   =>    |-  ( A  e.  V  ->  F : ( ( ~P A  i^i  Fin )  \  { (/) } ) -onto->
 ( fi `  A ) )
 
Theoremdcfi 7040* Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.)
 |-  ( ( A  e.  Fin  /\  A. x  e.  A DECID  ph )  -> DECID  A. x  e.  A  ph )
 
2.6.34  Supremum and infimum
 
Syntaxcsup 7041 Extend class notation to include supremum of class  A. Here  R is ordinarily a relation that strictly orders class  B. For example,  R could be 'less than' and  B could be the set of real numbers.
 class  sup ( A ,  B ,  R )
 
Syntaxcinf 7042 Extend class notation to include infimum of class  A. Here  R is ordinarily a relation that strictly orders class  B. For example,  R could be 'less than' and  B could be the set of real numbers.
 class inf ( A ,  B ,  R )
 
Definitiondf-sup 7043* Define the supremum of class  A. It is meaningful when 
R is a relation that strictly orders  B and when the supremum exists. (Contributed by NM, 22-May-1999.)
 |- 
 sup ( A ,  B ,  R )  =  U. { x  e.  B  |  ( A. y  e.  A  -.  x R y  /\  A. y  e.  B  (
 y R x  ->  E. z  e.  A  y R z ) ) }
 
Definitiondf-inf 7044 Define the infimum of class  A. It is meaningful when 
R is a relation that strictly orders 
B and when the infimum exists. For example,  R could be 'less than',  B could be the set of real numbers, and  A could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
 |- inf
 ( A ,  B ,  R )  =  sup ( A ,  B ,  `' R )
 
Theoremsupeq1 7045 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
 |-  ( B  =  C  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
 )
 
Theoremsupeq1d 7046 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R ) )
 
Theoremsupeq1i 7047 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  B  =  C   =>    |-  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
 
Theoremsupeq2 7048 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( B  =  C  ->  sup ( A ,  B ,  R )  =  sup ( A ,  C ,  R )
 )
 
Theoremsupeq3 7049 Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  ( R  =  S  ->  sup ( A ,  B ,  R )  =  sup ( A ,  B ,  S )
 )
 
Theoremsupeq123d 7050 Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  ( ph  ->  A  =  D )   &    |-  ( ph  ->  B  =  E )   &    |-  ( ph  ->  C  =  F )   =>    |-  ( ph  ->  sup ( A ,  B ,  C )  =  sup ( D ,  E ,  F ) )
 
Theoremnfsup 7051 Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x R   =>    |-  F/_ x sup ( A ,  B ,  R )
 
Theoremsupmoti 7052* Any class  B has at most one supremum in  A (where  R is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 8099) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ph  ->  E* x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
 
Theoremsupeuti 7053* A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by Jim Kingdon, 23-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  E! x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
 
Theoremsupval2ti 7054* Alternate expression for the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  (
 iota_ x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) ) )
 
Theoremeqsupti 7055* Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ph  ->  ( ( C  e.  A  /\  A. y  e.  B  -.  C R y  /\  A. y  e.  A  (
 y R C  ->  E. z  e.  B  y R z ) ) 
 ->  sup ( B ,  A ,  R )  =  C ) )
 
Theoremeqsuptid 7056* Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 24-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  -.  C R y )   &    |-  ( ( ph  /\  ( y  e.  A  /\  y R C ) )  ->  E. z  e.  B  y R z )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
 
Theoremsupclti 7057* A supremum belongs to its base class (closure law). See also supubti 7058 and suplubti 7059. (Contributed by Jim Kingdon, 24-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
 
Theoremsupubti 7058* A supremum is an upper bound. See also supclti 7057 and suplubti 7059.

This proof demonstrates how to expand an iota-based definition (df-iota 5215) using riotacl2 5887.

(Contributed by Jim Kingdon, 24-Nov-2021.)

 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  ( C  e.  B  ->  -. 
 sup ( B ,  A ,  R ) R C ) )
 
Theoremsuplubti 7059* A supremum is the least upper bound. See also supclti 7057 and supubti 7058. (Contributed by Jim Kingdon, 24-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  ( ( C  e.  A  /\  C R sup ( B ,  A ,  R ) )  ->  E. z  e.  B  C R z ) )
 
Theoremsuplub2ti 7060* Bidirectional form of suplubti 7059. (Contributed by Jim Kingdon, 17-Jan-2022.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   &    |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  B  C_  A )   =>    |-  ( ( ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R ) 
 <-> 
 E. z  e.  B  C R z ) )
 
Theoremsupelti 7061* Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  C  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   &    |-  ( ph  ->  C  C_  A )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  e.  C )
 
Theoremsup00 7062 The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
 |- 
 sup ( B ,  (/)
 ,  R )  =  (/)
 
Theoremsupmaxti 7063* The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  C  e.  B )   &    |-  (
 ( ph  /\  y  e.  B )  ->  -.  C R y )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
 
Theoremsupsnti 7064* The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  sup ( { B } ,  A ,  R )  =  B )
 
Theoremisotilem 7065* Lemma for isoti 7066. (Contributed by Jim Kingdon, 26-Nov-2021.)
 |-  ( F  Isom  R ,  S  ( A ,  B )  ->  ( A. x  e.  B  A. y  e.  B  ( x  =  y  <->  ( -.  x S y  /\  -.  y S x ) )  ->  A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) ) )
 
Theoremisoti 7066* An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)
 |-  ( F  Isom  R ,  S  ( A ,  B )  ->  ( A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) )  <->  A. u  e.  B  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) ) )
 
Theoremsupisolem 7067* Lemma for supisoti 7069. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  C 
 C_  A )   =>    |-  ( ( ph  /\  D  e.  A ) 
 ->  ( ( A. y  e.  C  -.  D R y  /\  A. y  e.  A  ( y R D  ->  E. z  e.  C  y R z ) )  <->  ( A. w  e.  ( F " C )  -.  ( F `  D ) S w 
 /\  A. w  e.  B  ( w S ( F `
  D )  ->  E. v  e.  ( F " C ) w S v ) ) ) )
 
Theoremsupisoex 7068* Lemma for supisoti 7069. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )   =>    |-  ( ph  ->  E. u  e.  B  ( A. w  e.  ( F " C )  -.  u S w 
 /\  A. w  e.  B  ( w S u  ->  E. v  e.  ( F " C ) w S v ) ) )
 
Theoremsupisoti 7069* Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
 |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )   &    |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A ) )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ph  ->  sup ( ( F " C ) ,  B ,  S )  =  ( F ` 
 sup ( C ,  A ,  R )
 ) )
 
Theoreminfeq1 7070 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( B  =  C  -> inf ( B ,  A ,  R )  = inf ( C ,  A ,  R ) )
 
Theoreminfeq1d 7071 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  -> inf ( B ,  A ,  R )  = inf ( C ,  A ,  R ) )
 
Theoreminfeq1i 7072 Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  B  =  C   =>    |- inf ( B ,  A ,  R )  = inf ( C ,  A ,  R )
 
Theoreminfeq2 7073 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( B  =  C  -> inf ( A ,  B ,  R )  = inf ( A ,  C ,  R ) )
 
Theoreminfeq3 7074 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( R  =  S  -> inf ( A ,  B ,  R )  = inf ( A ,  B ,  S ) )
 
Theoreminfeq123d 7075 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( ph  ->  A  =  D )   &    |-  ( ph  ->  B  =  E )   &    |-  ( ph  ->  C  =  F )   =>    |-  ( ph  -> inf ( A ,  B ,  C )  = inf ( D ,  E ,  F ) )
 
Theoremnfinf 7076 Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x R   =>    |-  F/_ xinf ( A ,  B ,  R )
 
Theoremcnvinfex 7077* Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.)
 |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x `' R y 
 /\  A. y  e.  A  ( y `' R x  ->  E. z  e.  B  y `' R z ) ) )
 
Theoremcnvti 7078* If a relation satisfies a condition corresponding to tightness of an apartness generated by an order, so does its converse. (Contributed by Jim Kingdon, 17-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u `' R v  /\  -.  v `' R u ) ) )
 
Theoremeqinfti 7079* Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ph  ->  ( ( C  e.  A  /\  A. y  e.  B  -.  y R C  /\  A. y  e.  A  ( C R y  ->  E. z  e.  B  z R y ) )  -> inf ( B ,  A ,  R )  =  C )
 )
 
Theoremeqinftid 7080* Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  -.  y R C )   &    |-  ( ( ph  /\  (
 y  e.  A  /\  C R y ) ) 
 ->  E. z  e.  B  z R y )   =>    |-  ( ph  -> inf ( B ,  A ,  R )  =  C )
 
Theoreminfvalti 7081* Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  -> inf ( B ,  A ,  R )  =  ( iota_ x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) ) )
 
Theoreminfclti 7082* An infimum belongs to its base class (closure law). See also inflbti 7083 and infglbti 7084. (Contributed by Jim Kingdon, 17-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  -> inf ( B ,  A ,  R )  e.  A )
 
Theoreminflbti 7083* An infimum is a lower bound. See also infclti 7082 and infglbti 7084. (Contributed by Jim Kingdon, 18-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  ->  ( C  e.  B  ->  -.  C Rinf ( B ,  A ,  R ) ) )
 
Theoreminfglbti 7084* An infimum is the greatest lower bound. See also infclti 7082 and inflbti 7083. (Contributed by Jim Kingdon, 18-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  ->  ( ( C  e.  A  /\ inf ( B ,  A ,  R ) R C )  ->  E. z  e.  B  z R C ) )
 
Theoreminfnlbti 7085* A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  ->  ( ( C  e.  A  /\  A. z  e.  B  -.  z R C ) 
 ->  -. inf ( B ,  A ,  R ) R C ) )
 
Theoreminfminti 7086* The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  C  e.  B )   &    |-  (
 ( ph  /\  y  e.  B )  ->  -.  y R C )   =>    |-  ( ph  -> inf ( B ,  A ,  R )  =  C )
 
Theoreminfmoti 7087* Any class  B has at most one infimum in  A (where  R is interpreted as 'less than'). (Contributed by Jim Kingdon, 18-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ph  ->  E* x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )
 
Theoreminfeuti 7088* An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  ->  E! x  e.  A  (
 A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )
 
Theoreminfsnti 7089* The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  -> inf ( { B } ,  A ,  R )  =  B )
 
Theoreminf00 7090 The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
 |- inf
 ( B ,  (/) ,  R )  =  (/)
 
Theoreminfisoti 7091* Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
 |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  C  z R y ) ) )   &    |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A ) )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ph  -> inf ( ( F " C ) ,  B ,  S )  =  ( F ` inf ( C ,  A ,  R ) ) )
 
Theoremsupex2g 7092 Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  C  ->  sup ( B ,  A ,  R )  e.  _V )
 
Theoreminfex2g 7093 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
 |-  ( A  e.  C  -> inf ( B ,  A ,  R )  e.  _V )
 
2.6.35  Ordinal isomorphism
 
Theoremordiso2 7094 Generalize ordiso 7095 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  Isom  _E 
 ,  _E  ( A ,  B )  /\  Ord 
 A  /\  Ord  B ) 
 ->  A  =  B )
 
Theoremordiso 7095* Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B 
 <-> 
 E. f  f  Isom  _E 
 ,  _E  ( A ,  B ) ) )
 
2.6.36  Disjoint union
 
2.6.36.1  Disjoint union
 
Syntaxcdju 7096 Extend class notation to include disjoint union of two classes.
 class  ( A B )
 
Definitiondf-dju 7097 Disjoint union of two classes. This is a way of creating a class which contains elements corresponding to each element of  A or  B, tagging each one with whether it came from  A or  B. (Contributed by Jim Kingdon, 20-Jun-2022.)
 |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B ) )
 
Theoremdjueq12 7098 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A C )  =  ( B D ) )
 
Theoremdjueq1 7099 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  ( A  =  B  ->  ( A C )  =  ( B C )
 )
 
Theoremdjueq2 7100 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  ( A  =  B  ->  ( C A )  =  ( C B )
 )
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