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Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremen1eqsnbi 6601 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 6602* A case where the antecedent of snexg 3992 is not needed. The class  { x  | 
ph } is from dcextest 4368. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
 |- 
 { { x  |  ph
 } }  e.  _V
 
2.6.30  Schroeder-Bernstein Theorem
 
Theoremsbthlem1 6603* Lemma for isbth 6613. (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 6604* Lemma for isbth 6613. (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 6605* Lemma for isbth 6613. (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 6606* Lemma for isbth 6613. (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 6607* Lemma for isbth 6613. (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 6608* Lemma for isbth 6613. (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 6609* Lemma for isbth 6613. (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 6610* Lemma for isbth 6613. (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 6611* Lemma for isbth 6613. (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 6612* Lemma for isbth 6613. (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 6613 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 6603 through sbthlemi10 6612; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 6612. 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 11343. (Contributed by NM, 8-Jun-1998.)
 |-  ( (EXMID 
 /\  ( A  ~<_  B  /\  B 
 ~<_  A ) )  ->  A  ~~  B )
 
2.6.31  Supremum and infimum
 
Syntaxcsup 6614 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 6615 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 6616* 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 6617 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 6618 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
 |-  ( B  =  C  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
 )
 
Theoremsupeq1d 6619 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R ) )
 
Theoremsupeq1i 6620 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  B  =  C   =>    |-  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
 
Theoremsupeq2 6621 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( B  =  C  ->  sup ( A ,  B ,  R )  =  sup ( A ,  C ,  R )
 )
 
Theoremsupeq3 6622 Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  ( R  =  S  ->  sup ( A ,  B ,  R )  =  sup ( A ,  B ,  S )
 )
 
Theoremsupeq123d 6623 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 6624 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 6625* 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 7502) 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 6626* 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 6627* 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 6628* 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 6629* 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 6630* A supremum belongs to its base class (closure law). See also supubti 6631 and suplubti 6632. (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 6631* A supremum is an upper bound. See also supclti 6630 and suplubti 6632.

This proof demonstrates how to expand an iota-based definition (df-iota 4943) using riotacl2 5576.

(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 6632* A supremum is the least upper bound. See also supclti 6630 and supubti 6631. (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 6633* Bidirectional form of suplubti 6632. (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 6634* 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 6635 The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
 |- 
 sup ( B ,  (/)
 ,  R )  =  (/)
 
Theoremsupmaxti 6636* 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 6637* 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 6638* Lemma for isoti 6639. (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 6639* 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 6640* Lemma for supisoti 6642. (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 6641* Lemma for supisoti 6642. (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 6642* 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 6643 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( B  =  C  -> inf ( B ,  A ,  R )  = inf ( C ,  A ,  R ) )
 
Theoreminfeq1d 6644 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  -> inf ( B ,  A ,  R )  = inf ( C ,  A ,  R ) )
 
Theoreminfeq1i 6645 Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  B  =  C   =>    |- inf ( B ,  A ,  R )  = inf ( C ,  A ,  R )
 
Theoreminfeq2 6646 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( B  =  C  -> inf ( A ,  B ,  R )  = inf ( A ,  C ,  R ) )
 
Theoreminfeq3 6647 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( R  =  S  -> inf ( A ,  B ,  R )  = inf ( A ,  B ,  S ) )
 
Theoreminfeq123d 6648 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 6649 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 6650* 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 6651* 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 6652* 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 6653* 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 6654* 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 6655* An infimum belongs to its base class (closure law). See also inflbti 6656 and infglbti 6657. (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 6656* An infimum is a lower bound. See also infclti 6655 and infglbti 6657. (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 6657* An infimum is the greatest lower bound. See also infclti 6655 and inflbti 6656. (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 6658* 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 6659* 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 6660* 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 6661* 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 6662* 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 6663 The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
 |- inf
 ( B ,  (/) ,  R )  =  (/)
 
Theoreminfisoti 6664* 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 ) ) )
 
2.6.32  Ordinal isomorphism
 
Theoremordiso2 6665 Generalize ordiso 6666 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  Isom  _E 
 ,  _E  ( A ,  B )  /\  Ord 
 A  /\  Ord  B ) 
 ->  A  =  B )
 
Theoremordiso 6666* 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 ) ) )
 
2.6.33  Disjoint union
 
2.6.33.1  Disjoint union
 
Syntaxcdju 6667 Extend class notation to include disjoint union of two classes.
 class  ( A B )
 
Definitiondf-dju 6668 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 6669 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A C )  =  ( B D ) )
 
Theoremdjueq1 6670 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  ( A  =  B  ->  ( A C )  =  ( B C )
 )
 
Theoremdjueq2 6671 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  ( A  =  B  ->  ( C A )  =  ( C B )
 )
 
Theoremnfdju 6672 Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A B )
 
Theoremdjuex 6673 The disjoint union of sets is a set. (Contributed by AV, 28-Jun-2022.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B )  e.  _V )
 
2.6.33.2  Left and right injections of a disjoint union

In this section, we define the left and right injections of a disjoint union and prove their main properties. These injections are restrictions of the "template" functions inl and inr, which appear in most applications in the form  (inl  |`  A ) and  (inr  |`  B ).

 
Syntaxcinl 6674 Extend class notation to include left injection of a disjoint union.
 class inl
 
Syntaxcinr 6675 Extend class notation to include right injection of a disjoint union.
 class inr
 
Definitiondf-inl 6676 Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
 |- inl 
 =  ( x  e. 
 _V  |->  <. (/) ,  x >. )
 
Definitiondf-inr 6677 Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
 |- inr 
 =  ( x  e. 
 _V  |->  <. 1o ,  x >. )
 
Theoremdjulclr 6678 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
 |-  ( C  e.  A  ->  ( (inl  |`  A ) `
  C )  e.  ( A B )
 )
 
Theoremdjurclr 6679 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
 |-  ( C  e.  B  ->  ( (inr  |`  B ) `
  C )  e.  ( A B )
 )
 
Theoremdjulcl 6680 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
 |-  ( C  e.  A  ->  (inl `  C )  e.  ( A B )
 )
 
Theoremdjurcl 6681 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
 |-  ( C  e.  B  ->  (inr `  C )  e.  ( A B )
 )
 
Theoremdjuf1olem 6682* Lemma for djulf1o 6687 and djurf1o 6688. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
 |-  X  e.  _V   &    |-  F  =  ( x  e.  A  |->  <. X ,  x >. )   =>    |-  F : A -1-1-onto-> ( { X }  X.  A )
 
Theoremdjuf1olemr 6683* Lemma for djulf1or 6685 and djurf1or 6686. Remark: maybe a version of this lemma with  F defined on  A and no restriction in the conclusion would be more usable. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
 |-  X  e.  _V   &    |-  F  =  ( x  e.  _V  |->  <. X ,  x >. )   =>    |-  ( F  |`  A ) : A -1-1-onto-> ( { X }  X.  A )
 
Theoremdjulclb 6684 Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
 |-  ( C  e.  V  ->  ( C  e.  A  <->  (inl `  C )  e.  ( A B ) ) )
 
Theoremdjulf1or 6685 The left injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
 |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
 
Theoremdjurf1or 6686 The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
 |-  (inr  |`  A ) : A -1-1-onto-> ( { 1o }  X.  A )
 
Theoremdjulf1o 6687 The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
 |- inl : _V
 -1-1-onto-> ( { (/) }  X.  _V )
 
Theoremdjurf1o 6688 The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
 |- inr : _V
 -1-1-onto-> ( { 1o }  X.  _V )
 
Theoreminresflem 6689* Lemma for inlresf1 6690 and inrresf1 6691. (Contributed by BJ, 4-Jul-2022.)
 |-  F : A -1-1-onto-> ( { X }  X.  A )   &    |-  ( x  e.  A  ->  ( F `  x )  e.  B )   =>    |-  F : A -1-1-> B
 
Theoreminlresf1 6690 The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
 |-  (inl  |`  A ) : A -1-1-> ( A B )
 
Theoreminrresf1 6691 The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
 |-  (inr  |`  B ) : B -1-1-> ( A B )
 
Theoremdjuinr 6692 The ranges of any left and right injections are disjoint. Remark: the restrictions seem not necessary, but the proof is not much longer than the proof of  |-  ( ran inl  i^i 
ran inr )  =  (/) (which is easily recovered from it, as in the proof of casefun 6713). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
 |-  ( ran  (inl  |`  A )  i^i  ran  (inr  |`  B ) )  =  (/)
 
Theoremdjuin 6693 The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.)
 |-  ( (inl " A )  i^i  (inr " B ) )  =  (/)
 
Theoremdjur 6694* A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  ( C  e.  ( A B )  ->  ( E. x  e.  A  C  =  (inl `  x )  \/  E. x  e.  B  C  =  (inr `  x ) ) )
 
Theoremdjuunr 6695 The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.)
 |-  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )  =  ( A B )
 
Theoremeldju 6696* Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
 |-  ( C  e.  ( A B )  <->  ( E. x  e.  A  C  =  ( (inl  |`  A ) `  x )  \/  E. x  e.  B  C  =  ( (inr  |`  B ) `  x ) ) )
 
Theoremdjuun 6697 The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.)
 |-  ( (inl " A )  u.  (inr " B ) )  =  ( A B )
 
2.6.33.3  Universal property of the disjoint union
 
Theoremdjuss 6698 A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
 |-  ( A B )  C_  ( { (/) ,  1o }  X.  ( A  u.  B ) )
 
Theoremeldju1st 6699 The first component of an element of a disjoint union is either  (/) or  1o. (Contributed by AV, 26-Jun-2022.)
 |-  ( X  e.  ( A B )  ->  (
 ( 1st `  X )  =  (/)  \/  ( 1st `  X )  =  1o ) )
 
Theoremeldju2ndl 6700 The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.)
 |-  ( ( X  e.  ( A B )  /\  ( 1st `  X )  =  (/) )  ->  ( 2nd `  X )  e.  A )
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