HomeHome Metamath Proof Explorer
Theorem List (p. 71 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21494)
  Hilbert Space Explorer  Hilbert Space Explorer
(21495-23017)
  Users' Mathboxes  Users' Mathboxes
(23018-31433)
 

Theorem List for Metamath Proof Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsdomen1 7001 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( A  ~<  C  <->  B  ~<  C ) )
 
Theoremsdomen2 7002 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( C  ~<  A  <->  C  ~<  B ) )
 
Theoremdomtriord 7003 Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
 
Theoremsdomel 7004 Strict dominance implies ordinal membership. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<  B 
 ->  A  e.  B ) )
 
Theoremsdomdif 7005 The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.)
 |-  ( A  ~<  B  ->  ( B  \  A )  =/=  (/) )
 
Theoremonsdominel 7006 An ordinal with more elements of some type is larger. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  ( A  i^i  C )  ~<  ( B  i^i  C ) )  ->  A  e.  B )
 
Theoremdomunsn 7007 Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~<  B  ->  ( A  u.  { C } )  ~<_  B )
 
Theoremfodomr 7008* There exists a mapping from a set onto any (non-empty) set that it dominates. (Contributed by NM, 23-Mar-2006.)
 |-  ( ( (/)  ~<  B  /\  B 
 ~<_  A )  ->  E. f  f : A -onto-> B )
 
Theorempwdom 7009 Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
 
Theoremcanth2 7010 Cantor's Theorem. No set is equinumerous to its power set. Specifically, any set has a cardinality (size) strictly less than the cardinality of its power set. For example, the cardinality of real numbers is the same as the cardinality of the power set of integers, so real numbers cannot be put into a one-to-one correspondence with integers. Theorem 23 of [Suppes] p. 97. For the function version, see canth 6288. (Contributed by NM, 7-Aug-1994.)
 |-  A  e.  _V   =>    |-  A  ~<  ~P A
 
Theoremcanth2g 7011 Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97. (Contributed by NM, 7-Nov-2003.)
 |-  ( A  e.  V  ->  A  ~<  ~P A )
 
Theorem2pwuninel 7012 The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by NM, 27-Jun-2008.)
 |- 
 -.  ~P ~P U. A  e.  A
 
Theorem2pwne 7013 No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.)
 |-  ( A  e.  V  ->  ~P ~P A  =/=  A )
 
Theoremdisjen 7014 A stronger form of pwuninel 6296. We can use pwuninel 6296, 2pwuninel 7012 to create one or two sets disjoint from a given set  A, but here we show that in fact such constructions exist for arbitrarily large disjoint extensions, which is to say that for any set  B we can construct a set  x that is equinumerous to it and disjoint from  A. (Contributed by Mario Carneiro, 7-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  i^i  ( B  X.  { ~P U. ran  A } ) )  =  (/)  /\  ( B  X.  { ~P U. ran  A } )  ~~  B ) )
 
Theoremdisjenex 7015* Existence version of disjen 7014. (Contributed by Mario Carneiro, 7-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x ( ( A  i^i  x )  =  (/)  /\  x  ~~  B ) )
 
Theoremdomss2 7016 A corollary of disjenex 7015. If  F is an injection from  A to  B then  G is a right inverse of  F from  B to a superset of  A. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  G  =  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )   =>    |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W )  ->  ( G : B
 -1-1-onto-> ran  G  /\  A  C_  ran 
 G  /\  ( G  o.  F )  =  (  _I  |`  A )
 ) )
 
Theoremdomssex2 7017* A corollary of disjenex 7015. If  F is an injection from  A to  B then there is a right inverse  g of  F from  B to a superset of  A. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W )  ->  E. g ( g : B -1-1-> _V  /\  ( g  o.  F )  =  (  _I  |`  A ) ) )
 
Theoremdomssex 7018* Weakening of domssex 7018 to forget the functions in favor of dominance and equinumerosity. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  ~<_  B  ->  E. x ( A  C_  x  /\  B  ~~  x ) )
 
2.4.31  Equinumerosity (cont.)
 
Theoremxpf1o 7019* Construct a bijection on a cross product given bijections on the factors. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  ( x  e.  A  |->  X ) : A -1-1-onto-> B )   &    |-  ( ph  ->  ( y  e.  C  |->  Y ) : C -1-1-onto-> D )   =>    |-  ( ph  ->  ( x  e.  A ,  y  e.  C  |->  <. X ,  Y >. ) : ( A  X.  C ) -1-1-onto-> ( B  X.  D ) )
 
Theoremxpen 7020 Equinumerosity law for cross product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
 
Theoremmapen 7021 Two set exponentiations are equinumerous when their bases and exponents are equinumerous. Theorem 6H(c) of [Enderton] p. 139. (Contributed by NM, 16-Dec-2003.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A  ^m  C )  ~~  ( B 
 ^m  D ) )
 
Theoremmapdom1 7022 Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
 |-  ( A  ~<_  B  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
 
Theoremmapxpen 7023 Equinumerosity law for double set exponentiation. Proposition 10.45 of [TakeutiZaring] p. 96. (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( ( A  ^m  B )  ^m  C ) 
 ~~  ( A  ^m  ( B  X.  C ) ) )
 
Theoremxpmapenlem 7024* Lemma for xpmapen 7025. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  =  ( z  e.  C  |->  ( 1st `  ( x `  z ) ) )   &    |-  R  =  ( z  e.  C  |->  ( 2nd `  ( x `  z ) ) )   &    |-  S  =  ( z  e.  C  |->  <.
 ( ( 1st `  y
 ) `  z ) ,  ( ( 2nd `  y
 ) `  z ) >. )   =>    |-  ( ( A  X.  B )  ^m  C ) 
 ~~  ( ( A 
 ^m  C )  X.  ( B  ^m  C ) )
 
Theoremxpmapen 7025 Equinumerosity law for set exponentiation of a cross product. Exercise 4.47 of [Mendelson] p. 255. (Contributed by NM, 23-Feb-2004.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  X.  B )  ^m  C )  ~~  ( ( A  ^m  C )  X.  ( B  ^m  C ) )
 
Theoremmapunen 7026 Equinumerosity law for set exponentiation of a disjoint union. Exercise 4.45 of [Mendelson] p. 255. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  /\  ( A  i^i  B )  =  (/) )  ->  ( C 
 ^m  ( A  u.  B ) )  ~~  ( ( C  ^m  A )  X.  ( C  ^m  B ) ) )
 
Theoremmap2xp 7027 A cardinal power with exponent 2 is equivalent to a cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( A  X.  A ) )
 
Theoremmapdom2 7028 Order-preserving property of set exponentiation. Theorem 6L(d) of [Enderton] p. 149. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) ) 
 ->  ( C  ^m  A ) 
 ~<_  ( C  ^m  B ) )
 
Theoremmapdom3 7029 Set exponentiation dominates the mantissa. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  B  =/=  (/) )  ->  A 
 ~<_  ( A  ^m  B ) )
 
Theorempwen 7030 If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 9-Apr-2015.)
 |-  ( A  ~~  B  ->  ~P A  ~~  ~P B )
 
Theoremssenen 7031* Equinumerosity of equinumerous subsets of a set. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  ~~  B  ->  { x  |  ( x  C_  A  /\  x  ~~  C ) }  ~~  { x  |  ( x  C_  B  /\  x  ~~  C ) }
 )
 
Theoremlimenpsi 7032 A limit ordinal is equinumerous to a proper subset of itself. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |- 
 Lim  A   =>    |-  ( A  e.  V  ->  A  ~~  ( A 
 \  { (/) } )
 )
 
Theoremlimensuci 7033 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
 |- 
 Lim  A   =>    |-  ( A  e.  V  ->  A  ~~  suc  A )
 
Theoremlimensuc 7034 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
 |-  ( ( A  e.  V  /\  Lim  A )  ->  A  ~~  suc  A )
 
Theoreminfensuc 7035 Any infinite ordinal is equinumerous to its successor. Exercise 7 of [TakeutiZaring] p. 88. Proved without the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 13-Jan-2013.)
 |-  ( ( A  e.  On  /\  om  C_  A )  ->  A  ~~  suc  A )
 
2.4.32  Pigeonhole Principle
 
Theoremphplem1 7036 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 7037 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 7038 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. (Contributed by NM, 26-May-1998.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremphplem4 7039 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 ) )
 
Theoremnneneq 7040 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 ) )
 
Theoremphp 7041 Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. 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 7036 through phplem4 7039, nneneq 7040, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  C.  A )  ->  -.  A  ~~  B )
 
Theoremphp2 7042 Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  C.  A )  ->  B  ~<  A )
 
Theoremphp3 7043 Corollary of Pigeonhole Principle. If  A is finite and 
B is a proper subset of  A, the  B is strictly less numerous than  A. Stronger version of Corollary 6C of [Enderton] p. 135. (Contributed by NM, 22-Aug-2008.)
 |-  ( ( A  e.  Fin  /\  B  C.  A ) 
 ->  B  ~<  A )
 
Theoremphp4 7044 Corollary of the Pigeonhole Principle php 7041: a natural number is strictly dominated by its successor. (Contributed by NM, 26-Jul-2004.)
 |-  ( A  e.  om  ->  A  ~<  suc  A )
 
Theoremphp5 7045 Corollary of the Pigeonhole Principle php 7041: 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 )
 
2.4.33  Finite sets
 
Theoremonomeneq 7046 An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.)
 |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  ~~  B 
 <->  A  =  B ) )
 
Theoremonfin 7047 An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92. (Contributed by NM, 26-Jul-2004.)
 |-  ( A  e.  On  ->  ( A  e.  Fin  <->  A  e.  om ) )
 
Theoremonfin2 7048 A set is a natural number iff it is a finite ordinal. (Contributed by Mario Carneiro, 22-Jan-2013.)
 |- 
 om  =  ( On 
 i^i  Fin )
 
Theoremnnfi 7049 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  ( A  e.  om  ->  A  e.  Fin )
 
Theoremnndomo 7050 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 ) )
 
Theoremnnsdomo 7051 Cardinal ordering agrees with natural number ordering. (Contributed by NM, 17-Jun-1998.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ~<  B  <->  A  C.  B ) )
 
TheoremomsucdomOLD 7052 Strict dominance of natural numbers is the same as dominance over the successor of the smaller. (Contributed by NM, 25-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
 
Theoremsucdom2 7053 Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
 |-  ( A  ~<  B  ->  suc 
 A  ~<_  B )
 
Theoremsucdom 7054 Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.)
 |-  ( A  e.  om  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
 
TheoremsucdomiOLD 7055 Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 7054. (Contributed by NM, 26-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  om 
 /\  B  e.  C )  ->  ( suc  A  ~<_  B  ->  A  ~<  B ) )
 
Theorem0sdom1dom 7056 Strict dominance over zero is the same as dominance over one. (Contributed by NM, 28-Sep-2004.)
 |-  ( (/)  ~<  A  <->  1o  ~<_  A )
 
Theorem1sdom2 7057 Ordinal 1 is strictly dominated by ordinal 2. (Contributed by NM, 4-Apr-2007.)
 |- 
 1o  ~<  2o
 
Theoremsdom1 7058 A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( A  ~<  1o  <->  A  =  (/) )
 
Theoremmodom 7059 Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E* x ph  <->  { x  |  ph }  ~<_  1o )
 
Theoremmodom2 7060* Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E* x  x  e.  A  <->  A  ~<_  1o )
 
Theorem1sdom 7061* A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 6929.) (Contributed by Mario Carneiro, 12-Jan-2013.)
 |-  ( A  e.  V  ->  ( 1o  ~<  A  <->  E. x  e.  A  E. y  e.  A  -.  x  =  y
 ) )
 
TheoremfisucdomOLD 7062 Strict dominance of a finite set over a natural number is the same as dominance over its successor. (Contributed by NM, 26-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  om 
 /\  B  e.  Fin )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
 
Theoremunxpdomlem1 7063* Lemma for unxpdom 7066. (Trivial substitution proof.) (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  F  =  ( x  e.  ( a  u.  b )  |->  G )   &    |-  G  =  if ( x  e.  a ,  <. x ,  if ( x  =  m ,  t ,  s ) >. ,  <. if ( x  =  t ,  n ,  m ) ,  x >. )   =>    |-  ( z  e.  (
 a  u.  b ) 
 ->  ( F `  z
 )  =  if (
 z  e.  a , 
 <. z ,  if (
 z  =  m ,  t ,  s ) >. ,  <. if ( z  =  t ,  n ,  m ) ,  z >. ) )
 
Theoremunxpdomlem2 7064* Lemma for unxpdom 7066. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  F  =  ( x  e.  ( a  u.  b )  |->  G )   &    |-  G  =  if ( x  e.  a ,  <. x ,  if ( x  =  m ,  t ,  s ) >. ,  <. if ( x  =  t ,  n ,  m ) ,  x >. )   &    |-  ( ph  ->  w  e.  ( a  u.  b ) )   &    |-  ( ph  ->  -.  m  =  n )   &    |-  ( ph  ->  -.  s  =  t )   =>    |-  ( ( ph  /\  (
 z  e.  a  /\  -.  w  e.  a ) )  ->  -.  ( F `  z )  =  ( F `  w ) )
 
Theoremunxpdomlem3 7065* Lemma for unxpdom 7066. (Contributed by Mario Carneiro, 13-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  F  =  ( x  e.  ( a  u.  b )  |->  G )   &    |-  G  =  if ( x  e.  a ,  <. x ,  if ( x  =  m ,  t ,  s ) >. ,  <. if ( x  =  t ,  n ,  m ) ,  x >. )   =>    |-  ( ( 1o  ~<  a 
 /\  1o  ~<  b ) 
 ->  ( a  u.  b
 )  ~<_  ( a  X.  b ) )
 
Theoremunxpdom 7066 Cross product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( 1o  ~<  A 
 /\  1o  ~<  B ) 
 ->  ( A  u.  B ) 
 ~<_  ( A  X.  B ) )
 
Theoremunxpdom2 7067 Corollary of unxpdom 7066. (Contributed by NM, 16-Sep-2004.)
 |-  ( ( 1o  ~<  A 
 /\  B  ~<_  A ) 
 ->  ( A  u.  B ) 
 ~<_  ( A  X.  A ) )
 
Theoremsucxpdom 7068 Cross product dominates successor for set with cardinality greater than 1. Proposition 10.38 of [TakeutiZaring] p. 93 (but generalized to arbitrary sets, not just ordinals). (Contributed by NM, 3-Sep-2004.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
 |-  ( 1o  ~<  A  ->  suc 
 A  ~<_  ( A  X.  A ) )
 
Theorempssinf 7069 A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
 |-  ( ( A  C.  B  /\  A  ~~  B )  ->  -.  B  e.  Fin )
 
Theoremfisseneq 7070 A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
 |-  ( ( B  e.  Fin  /\  A  C_  B  /\  A  ~~  B )  ->  A  =  B )
 
Theoremominf 7071 The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
 |- 
 -.  om  e.  Fin
 
Theoremisinf 7072* Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by Mario Carneiro, 15-Jan-2013.)
 |-  ( -.  A  e.  Fin 
 ->  A. n  e.  om  E. x ( x  C_  A  /\  x  ~~  n ) )
 
Theoremfineqvlem 7073 Lemma for fineqv 7074. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  om  ~<_  ~P ~P A )
 
Theoremfineqv 7074 If the Axiom of Infinity is denied, then all sets are finite (which implies the Axiom of Choice). (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 3-Jan-2015.)
 |-  ( -.  om  e.  _V  <->  Fin 
 =  _V )
 
Theoremenfi 7075 Equinmerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
 |-  ( A  ~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
 
Theoremenfii 7076 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( B  e.  Fin  /\  A  ~~  B ) 
 ->  A  e.  Fin )
 
Theorempssnn 7077* A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137. (Contributed by NM, 22-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  C.  A )  ->  E. x  e.  A  B  ~~  x )
 
Theoremssnnfi 7078 A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.)
 |-  ( ( A  e.  om 
 /\  B  C_  A )  ->  B  e.  Fin )
 
Theoremssfi 7079 A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138. (Contributed by NM, 24-Jun-1998.)
 |-  ( ( A  e.  Fin  /\  B  C_  A )  ->  B  e.  Fin )
 
Theoremdomfi 7080 A set dominated by a finite set is finite. (Contributed by NM, 23-Mar-2006.) (Revised by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( A  e.  Fin  /\  B  ~<_  A )  ->  B  e.  Fin )
 
Theoremxpfir 7081 The components of a non-empty finite cross product are finite. (Contributed by Paul Chapman, 11-Apr-2009.) (Proof shortened by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( ( A  X.  B )  e. 
 Fin  /\  ( A  X.  B )  =/=  (/) )  ->  ( A  e.  Fin  /\  B  e.  Fin )
 )
 
Theoremf1finf1o 7082 Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.)
 |-  ( ( A  ~~  B  /\  B  e.  Fin )  ->  ( F : A -1-1-> B  <->  F : A -1-1-onto-> B ) )
 
Theorem0fin 7083 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
 |-  (/)  e.  Fin
 
Theoremen1eqsn 7084 A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
 |-  ( ( A  e.  B  /\  B  ~~  1o )  ->  B  =  { A } )
 
Theoremdiffi 7085 If  A is finite,  ( A 
\  B ) is finite. (Contributed by FL, 3-Aug-2009.)
 |-  ( A  e.  Fin  ->  ( A  \  B )  e.  Fin )
 
Theoremdif1enOLD 7086 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.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( ( M  e.  om  /\  A  ~~ 
 suc  M  /\  X  e.  A )  ->  ( A 
 \  { X }
 )  ~~  M )
 
Theoremdif1en 7087 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 )
 
Theoremenp1ilem 7088 Lemma for uses of enp1i 7089. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  T  =  ( { x }  u.  S )   =>    |-  ( x  e.  A  ->  ( ( A  \  { x } )  =  S  ->  A  =  T ) )
 
Theoremenp1i 7089* Proof induction for en2i 6895 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  M  e.  om   &    |-  N  =  suc  M   &    |-  ( ( A 
 \  { x }
 )  ~~  M  ->  ph )   &    |-  ( x  e.  A  ->  ( ph  ->  ps ) )   =>    |-  ( A  ~~  N  ->  E. x ps )
 
Theoremen2 7090* A set equinumerous to ordinal 2 is a pair. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( A  ~~  2o  ->  E. x E. y  A  =  { x ,  y } )
 
Theoremen3 7091* A set equinumerous to ordinal 3 is a triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( A  ~~  3o  ->  E. x E. y E. z  A  =  { x ,  y ,  z } )
 
Theoremen4 7092* A set equinumerous to ordinal 4 is a quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( A  ~~  4o  ->  E. x E. y E. z E. w  A  =  ( { x ,  y }  u.  { z ,  w } ) )
 
Theoremfindcard 7093* 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 7094* 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 7095* Variation of findcard2 7094 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 )
 
Theoremfindcard3 7096* Schema for strong induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on any proper subset. The result is then proven to be true for all finite sets. (Contributed by Mario Carneiro, 13-Dec-2013.)
 |-  ( x  =  y 
 ->  ( ph  <->  ch ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  (
 y  e.  Fin  ->  (
 A. x ( x 
 C.  y  ->  ph )  ->  ch ) )   =>    |-  ( A  e.  Fin 
 ->  ta )
 
Theoremac6sfi 7097* A version of ac6s 8107 for finite sets. (Contributed by Jeffrey 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 ) )
 
Theoremfrfi 7098 A partial order is well-founded on a finite set. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Po  A  /\  A  e.  Fin )  ->  R  Fr  A )
 
Theoremfimax2g 7099* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremfimaxg 7100* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  y R x ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31433
  Copyright terms: Public domain < Previous  Next >