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Theorem List for Intuitionistic Logic Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremendomtr 6301 Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
 |-  ( ( A  ~~  B  /\  B  ~<_  C ) 
 ->  A  ~<_  C )
 
Theoremdomentr 6302 Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
 |-  ( ( A  ~<_  B  /\  B  ~~  C )  ->  A 
 ~<_  C )
 
Theoremf1imaeng 6303 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  ( F " C )  ~~  C )
 
Theoremf1imaen2g 6304 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6305 does not need ax-setind 4290.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( ( F : A -1-1-> B  /\  B  e.  V )  /\  ( C  C_  A  /\  C  e.  V ) )  ->  ( F " C )  ~~  C )
 
Theoremf1imaen 6305 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)
 |-  C  e.  _V   =>    |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F " C ) 
 ~~  C )
 
Theoremen0 6306 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
 |-  ( A  ~~  (/)  <->  A  =  (/) )
 
Theoremensn1 6307 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
 |-  A  e.  _V   =>    |-  { A }  ~~  1o
 
Theoremensn1g 6308 A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
 |-  ( A  e.  V  ->  { A }  ~~  1o )
 
Theoremenpr1g 6309  { A ,  A } has only one element. (Contributed by FL, 15-Feb-2010.)
 |-  ( A  e.  V  ->  { A ,  A }  ~~  1o )
 
Theoremen1 6310* A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
 |-  ( A  ~~  1o  <->  E. x  A  =  { x } )
 
Theoremen1bg 6311 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)
 |-  ( A  e.  V  ->  ( A  ~~  1o  <->  A  =  { U. A }
 ) )
 
Theoremreuen1 6312* Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E! x  e.  A  ph  <->  { x  e.  A  |  ph }  ~~  1o )
 
Theoremeuen1 6313 Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E! x ph  <->  { x  |  ph }  ~~  1o )
 
Theoremeuen1b 6314* Two ways to express " A has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  ( A  ~~  1o  <->  E! x  x  e.  A )
 
Theoremen1uniel 6315 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  ( S  ~~  1o  ->  U. S  e.  S )
 
Theorem2dom 6316* A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
 |-  ( 2o  ~<_  A  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y )
 
Theoremfundmen 6317 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  F  e.  _V   =>    |-  ( Fun  F  ->  dom  F  ~~  F )
 
Theoremfundmeng 6318 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
 |-  ( ( F  e.  V  /\  Fun  F )  ->  dom  F  ~~  F )
 
Theoremcnven 6319 A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( ( Rel  A  /\  A  e.  V ) 
 ->  A  ~~  `' A )
 
Theoremfndmeng 6320 A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( F  Fn  A  /\  A  e.  C )  ->  A  ~~  F )
 
Theoremen2sn 6321 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  { A }  ~~  { B } )
 
Theoremsnfig 6322 A singleton is finite. (Contributed by Jim Kingdon, 13-Apr-2020.)
 |-  ( A  e.  V  ->  { A }  e.  Fin )
 
Theoremfiprc 6323 The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
 |- 
 Fin  e/  _V
 
Theoremunen 6324 Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( ( A 
 ~~  B  /\  C  ~~  D )  /\  (
 ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( A  u.  C )  ~~  ( B  u.  D ) )
 
Theoremenm 6325* A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
 |-  ( ( A  ~~  B  /\  E. x  x  e.  A )  ->  E. y  y  e.  B )
 
Theoremxpsnen 6326 A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  X.  { B } )  ~~  A
 
Theoremxpsneng 6327 A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A )
 
Theoremxp1en 6328 One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  V  ->  ( A  X.  1o )  ~~  A )
 
Theoremendisj 6329* Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 E. x E. y
 ( ( x  ~~  A  /\  y  ~~  B )  /\  ( x  i^i  y )  =  (/) )
 
Theoremxpcomf1o 6330* The canonical bijection from  ( A  X.  B
) to  ( B  X.  A ). (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  F  =  ( x  e.  ( A  X.  B )  |->  U. `' { x } )   =>    |-  F : ( A  X.  B ) -1-1-onto-> ( B  X.  A )
 
Theoremxpcomco 6331* Composition with the bijection of xpcomf1o 6330 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  F  =  ( x  e.  ( A  X.  B )  |->  U. `' { x } )   &    |-  G  =  ( y  e.  B ,  z  e.  A  |->  C )   =>    |-  ( G  o.  F )  =  ( z  e.  A ,  y  e.  B  |->  C )
 
Theoremxpcomen 6332 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  X.  B )  ~~  ( B  X.  A )
 
Theoremxpcomeng 6333 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B )  ~~  ( B  X.  A ) )
 
Theoremxpsnen2g 6334 A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  B )
 
Theoremxpassen 6335 Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  X.  B )  X.  C )  ~~  ( A  X.  ( B  X.  C ) )
 
Theoremxpdom2 6336 Dominance law for Cartesian product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  C  e.  _V   =>    |-  ( A  ~<_  B  ->  ( C  X.  A )  ~<_  ( C  X.  B ) )
 
Theoremxpdom2g 6337 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( C  e.  V  /\  A  ~<_  B ) 
 ->  ( C  X.  A ) 
 ~<_  ( C  X.  B ) )
 
Theoremxpdom1g 6338 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( C  e.  V  /\  A  ~<_  B ) 
 ->  ( A  X.  C ) 
 ~<_  ( B  X.  C ) )
 
Theoremxpdom3m 6339* A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  E. x  x  e.  B )  ->  A  ~<_  ( A  X.  B ) )
 
Theoremxpdom1 6340 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.)
 |-  C  e.  _V   =>    |-  ( A  ~<_  B  ->  ( A  X.  C )  ~<_  ( B  X.  C ) )
 
Theoremfopwdom 6341 Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  e.  _V 
 /\  F : A -onto-> B )  ->  ~P B  ~<_  ~P A )
 
Theoremenen1 6342 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( A  ~~  C  <->  B 
 ~~  C ) )
 
Theoremenen2 6343 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( C  ~~  A  <->  C 
 ~~  B ) )
 
Theoremdomen1 6344 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( A  ~<_  C  <->  B  ~<_  C )
 )
 
Theoremdomen2 6345 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B )
 )
 
2.6.26  Pigeonhole Principle
 
Theoremphplem1 6346 Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element. (Contributed by NM, 25-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  e.  A )  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc  A  \  { B } )
 )
 
Theoremphplem2 6347 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremphplem3 6348 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 6350. (Contributed by NM, 26-May-1998.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremphplem4 6349 Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( suc  A  ~~ 
 suc  B  ->  A  ~~  B ) )
 
Theoremphplem3g 6350 A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6348 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( ( A  e.  om 
 /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremnneneq 6351 Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ~~  B 
 <->  A  =  B ) )
 
Theoremphp5 6352 A natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.)
 |-  ( A  e.  om  ->  -.  A  ~~  suc  A )
 
Theoremsnnen2og 6353 A singleton  { A } is never equinumerous with the ordinal number 2. If  A is a proper class, see snnen2oprc 6354. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( A  e.  V  ->  -.  { A }  ~~  2o )
 
Theoremsnnen2oprc 6354 A singleton  { A } is never equinumerous with the ordinal number 2. If  A is a set, see snnen2og 6353. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( -.  A  e.  _V 
 ->  -.  { A }  ~~  2o )
 
Theoremphplem4dom 6355 Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( suc  A  ~<_  suc  B  ->  A  ~<_  B ) )
 
Theoremphp5dom 6356 A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( A  e.  om  ->  -.  suc  A  ~<_  A )
 
Theoremnndomo 6357 Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ~<_  B  <->  A  C_  B ) )
 
Theoremphpm 6358* Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. By "proper subset" here we mean that there is an element which is in the natural number and not in the subset, or in symbols  E. x x  e.  ( A  \  B
) (which is stronger than not being equal in the absence of excluded middle). Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 6346 through phplem4 6349, nneneq 6351, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  C_  A  /\  E. x  x  e.  ( A  \  B ) )  ->  -.  A  ~~  B )
 
Theoremphpelm 6359 Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)
 |-  ( ( A  e.  om 
 /\  B  e.  A )  ->  -.  A  ~~  B )
 
Theoremphplem4on 6360 Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)
 |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( suc  A  ~~ 
 suc  B  ->  A  ~~  B ) )
 
2.6.27  Finite sets
 
Theoremfidceq 6361 Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that  { B ,  C } is finite would require showing it is equinumerous to  1o or to  2o but to show that you'd need to know  B  =  C or  -.  B  =  C, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  -> DECID  B  =  C )
 
Theoremfidifsnen 6362 All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)
 |-  ( ( X  e.  Fin  /\  A  e.  X  /\  B  e.  X )  ->  ( X  \  { A } )  ~~  ( X  \  { B }
 ) )
 
Theoremfidifsnid 6363 If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3538 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  A ) 
 ->  ( ( A  \  { B } )  u. 
 { B } )  =  A )
 
Theoremnnfi 6364 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  ( A  e.  om  ->  A  e.  Fin )
 
Theoremenfi 6365 Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
 |-  ( A  ~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
 
Theoremenfii 6366 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( B  e.  Fin  /\  A  ~~  B ) 
 ->  A  e.  Fin )
 
Theoremssfiexmid 6367* If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.)
 |- 
 A. x A. y
 ( ( x  e. 
 Fin  /\  y  C_  x )  ->  y  e.  Fin )   =>    |-  ( ph  \/  -.  ph )
 
Theoremdif1en 6368 If a set  A is equinumerous to the successor of a natural number  M, then  A with an element removed is equinumerous to  M. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)
 |-  ( ( M  e.  om 
 /\  A  ~~  suc  M 
 /\  X  e.  A )  ->  ( A  \  { X } )  ~~  M )
 
Theoremfiunsnnn 6369 Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.)
 |-  ( ( ( A  e.  Fin  /\  B  e.  ( _V  \  A ) )  /\  ( N  e.  om  /\  A  ~~  N ) )  ->  ( A  u.  { B } )  ~~  suc  N )
 
Theoremphp5fin 6370 A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  ( _V  \  A ) )  ->  -.  A  ~~  ( A  u.  { B }
 ) )
 
Theoremfisbth 6371 Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
 |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  ~~  B )
 
Theorem0fin 6372 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
 |-  (/)  e.  Fin
 
Theoremfin0 6373* A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
 |-  ( A  e.  Fin  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
 
Theoremfin0or 6374* A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
 |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  E. x  x  e.  A ) )
 
Theoremdiffitest 6375* If subtracting any set from a finite set gives a finite set, any proposition of the form  -.  ph is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove  A  e.  Fin  ->  ( A  \  B
)  e.  Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
 |- 
 A. a  e.  Fin  A. b ( a  \  b )  e.  Fin   =>    |-  ( -.  ph  \/  -.  -.  ph )
 
Theoremfindcard 6376* Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  ( y  \  { z } )  ->  ( ph  <->  ch ) )   &    |-  ( x  =  y  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  Fin  ->  (
 A. z  e.  y  ch  ->  th ) )   =>    |-  ( A  e.  Fin 
 ->  ta )
 
Theoremfindcard2 6377* Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  Fin  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  Fin  ->  ta )
 
Theoremfindcard2s 6378* Variation of findcard2 6377 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 ( y  e.  Fin  /\ 
 -.  z  e.  y
 )  ->  ( ch  ->  th ) )   =>    |-  ( A  e.  Fin 
 ->  ta )
 
Theoremfindcard2d 6379* Deduction version of findcard2 6377. If you also need  y  e.  Fin (which doesn't come for free due to ssfiexmid 6367), use findcard2sd 6380 instead. (Contributed by SO, 16-Jul-2018.)
 |-  ( x  =  (/)  ->  ( ps  <->  ch ) )   &    |-  ( x  =  y  ->  ( ps  <->  th ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ps  <->  ta ) )   &    |-  ( x  =  A  ->  ( ps  <->  et ) )   &    |-  ( ph  ->  ch )   &    |-  ( ( ph  /\  ( y  C_  A  /\  z  e.  ( A  \  y ) ) )  ->  ( th  ->  ta ) )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  et )
 
Theoremfindcard2sd 6380* Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)
 |-  ( x  =  (/)  ->  ( ps  <->  ch ) )   &    |-  ( x  =  y  ->  ( ps  <->  th ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ps  <->  ta ) )   &    |-  ( x  =  A  ->  ( ps  <->  et ) )   &    |-  ( ph  ->  ch )   &    |-  ( ( (
 ph  /\  y  e.  Fin )  /\  ( y 
 C_  A  /\  z  e.  ( A  \  y
 ) ) )  ->  ( th  ->  ta )
 )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  et )
 
Theoremdiffisn 6381 Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  A ) 
 ->  ( A  \  { B } )  e.  Fin )
 
Theoremdiffifi 6382 Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  B  C_  A )  ->  ( A  \  B )  e.  Fin )
 
Theoremac6sfi 6383* Existence of a choice function for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( y  =  ( f `  x ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  Fin  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremfientri3 6384 Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  ~<_  B  \/  B 
 ~<_  A ) )
 
Theoremnnwetri 6385* A natural number is well-ordered by 
_E. More specifically, this order both satisfies  We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
 |-  ( A  e.  om  ->  (  _E  We  A  /\  A. x  e.  A  A. y  e.  A  ( x  _E  y  \/  x  =  y  \/  y  _E  x ) ) )
 
Theoremonunsnss 6386 Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
 |-  ( ( B  e.  V  /\  ( A  u.  { B } )  e. 
 On )  ->  B  C_  A )
 
Theoremsnon0 6387 An ordinal which is a singleton is  { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
 |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  A  =  (/) )
 
2.6.28  Supremum and infimum
 
Syntaxcsup 6388 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 6389 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 6390* 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 6391 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 6392 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
 |-  ( B  =  C  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
 )
 
Theoremsupeq1d 6393 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R ) )
 
Theoremsupeq1i 6394 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  B  =  C   =>    |-  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
 
Theoremsupeq2 6395 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( B  =  C  ->  sup ( A ,  B ,  R )  =  sup ( A ,  C ,  R )
 )
 
Theoremsupeq3 6396 Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  ( R  =  S  ->  sup ( A ,  B ,  R )  =  sup ( A ,  B ,  S )
 )
 
Theoremsupeq123d 6397 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 6398 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 6399* 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 7157) 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 6400* 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 ) ) )
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