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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3pm3.2ni 23401 Triple negated disjuntion introduction. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  -.  ph   &    |-  -. 
 ps   &    |- 
 -.  ch   =>    |- 
 -.  ( ph  \/  ps 
 \/  ch )
 
Theorem3jaodd 23402 Double deduction form of 3jaoi 1250. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  et ) ) )   &    |-  ( ph  ->  ( ps  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch 
 \/  th  \/  ta )  ->  et ) ) )
 
Theorem3orit 23403 Closed form of 3ori 1247, (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  (
 ( ph  \/  ps  \/  ch )  <->  ( ( -.  ph  /\  -.  ps )  ->  ch ) )
 
Theorem3mix1d 23404 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch  \/  th ) )
 
Theorem3mix2d 23405 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps  \/  th ) )
 
Theorem3mix3d 23406 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  th  \/  ps ) )
 
Theorembiimpexp 23407 A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)
 |-  (
 ( ( ph  <->  ps )  ->  ch )  <->  ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  ch )
 ) )
 
Theorem3orel13 23408 Elimination of two disjuncts in a triple disjunction. (Contributed by Scott Fenton, 9-Jun-2011.)
 |-  (
 ( -.  ph  /\  -.  ch )  ->  ( ( ph  \/  ps  \/  ch )  ->  ps ) )
 
16.7.4  Misc. Useful Theorems
 
Theoremnepss 23409 Two classes are inequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)
 |-  ( A  =/=  B  <->  ( ( A  i^i  B )  C.  A  \/  ( A  i^i  B )  C.  B ) )
 
Theorem3ccased 23410 Triple disjunction form of ccased 918. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( ph  ->  ( ( ch 
 /\  et )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( ch  /\  ze )  ->  ps ) )   &    |-  ( ph  ->  ( ( ch 
 /\  si )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( th  /\  et )  ->  ps ) )   &    |-  ( ph  ->  ( ( th  /\ 
 ze )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( th  /\  si )  ->  ps ) )   &    |-  ( ph  ->  ( ( ta 
 /\  et )  ->  ps )
 )   &    |-  ( ph  ->  (
 ( ta  /\  ze )  ->  ps ) )   &    |-  ( ph  ->  ( ( ta 
 /\  si )  ->  ps )
 )   =>    |-  ( ph  ->  (
 ( ( ch  \/  th 
 \/  ta )  /\  ( et  \/  ze  \/  si ) )  ->  ps )
 )
 
Theoremdfso3 23411* Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)
 |-  ( R  Or  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z )  /\  ( x R y  \/  x  =  y  \/  y R x ) ) )
 
Theorembrtpid1 23412 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { <. A ,  B >. ,  C ,  D } B
 
Theorembrtpid2 23413 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { C ,  <. A ,  B >. ,  D } B
 
Theorembrtpid3 23414 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
 |-  A { C ,  D ,  <. A ,  B >. } B
 
16.7.5  Properties of reals and complexes
 
Theoremsqdivzi 23415 Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  (
 ( A  /  B ) ^ 2 )  =  ( ( A ^
 2 )  /  ( B ^ 2 ) ) )
 
Theoremdivelunit 23416 A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  A  <_  B ) )
 
Theorempm2.61iine 23417 Equality version of pm2.61ii 159. (Contributed by Scott Fenton, 13-Jun-2013.)
 |-  (
 ( A  =/=  C  /\  B  =/=  D ) 
 ->  ph )   &    |-  ( A  =  C  ->  ph )   &    |-  ( B  =  D  ->  ph )   =>    |-  ph
 
Theoremdedekind 23418* The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 8748 with appropriate adjustments, states that, if  A completely preceeds  B, then there is some number separating the two of them. (Contributed by Scott Fenton, 13-Jun-2013.)
 |-  (
 ( A  C_  RR  /\  B  C_  RR  /\  A. x  e.  A  A. y  e.  B  x  <  y
 )  ->  E. z  e.  RR  A. x  e.  A  A. y  e.  B  ( x  <_  z  /\  z  <_  y
 ) )
 
Theoremdedekindle 23419* The Dedekind cut theorem, with the hypothesis weakened to only require non-strict less than. (Contributed by Scott Fenton, 2-Jul-2013.)
 |-  (
 ( A  C_  RR  /\  B  C_  RR  /\  A. x  e.  A  A. y  e.  B  x  <_  y
 )  ->  E. z  e.  RR  A. x  e.  A  A. y  e.  B  ( x  <_  z  /\  z  <_  y
 ) )
 
Theoremmulcan1g 23420 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  =  ( A  x.  C )  <->  ( A  =  0  \/  B  =  C ) ) )
 
Theoremmulcan2g 23421 A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  C )  =  ( B  x.  C )  <->  ( A  =  B  \/  C  =  0 ) ) )
 
Theoremmulge0b 23422 A condition for multiplication to be non-negative. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  ( A  x.  B )  <->  ( ( A 
 <_  0  /\  B  <_  0 )  \/  ( 0 
 <_  A  /\  0  <_  B ) ) ) )
 
Theoremmulle0b 23423 A condition for multiplication to be non-positive. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  <_  0  <->  ( ( A 
 <_  0  /\  0  <_  B )  \/  (
 0  <_  A  /\  B  <_  0 ) ) ) )
 
Theoremmulsuble0b 23424 A condition for multiplication of subtraction to be non-positive. (Contributed by Scott Fenton, 25-Jun-2013.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( ( A  -  B )  x.  ( C  -  B ) )  <_  0  <->  ( ( A 
 <_  B  /\  B  <_  C )  \/  ( C 
 <_  B  /\  B  <_  A ) ) ) )
 
Theoremrelin01 23425 An interval law for less than or equal. (Contributed by Scott Fenton, 27-Jun-2013.)
 |-  ( A  e.  RR  ->  ( A  <_  0  \/  ( 0  <_  A  /\  A  <_  1 )  \/  1  <_  A ) )
 
Theoremsubdivcomb1 23426 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( (
 ( C  x.  A )  -  B )  /  C )  =  ( A  -  ( B  /  C ) ) )
 
Theoremsubdivcomb2 23427 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  -  ( C  x.  B ) )  /  C )  =  (
 ( A  /  C )  -  B ) )
 
Theoremsubeqrev 23428 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  =  ( C  -  D )  <->  ( B  -  A )  =  ( D  -  C ) ) )
 
Theoremfznatpl1 23429 Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)
 |-  (
 ( N  e.  NN  /\  I  e.  ( 1
 ... ( N  -  1 ) ) ) 
 ->  ( I  +  1 )  e.  ( 1
 ... N ) )
 
Theoremsupfz 23430 The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  sup (
 ( M ... N ) ,  ZZ ,  <  )  =  N )
 
Theoreminffz 23431 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  sup (
 ( M ... N ) ,  ZZ ,  `'  <  )  =  M )
 
Theorembcnm1 23432 The binomial coefficent of  ( N  -  1 ) is  N. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  ( N  -  1 ) )  =  N )
 
Theoremfz0n 23433 The sequence  ( 0 ... ( N  -  1 ) ) is empty iff  N is zero. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  (
 ( 0 ... ( N  -  1 ) )  =  (/)  <->  N  =  0
 ) )
 
Theorem4bc3eq4 23434 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
 |-  (
 4  _C  3 )  =  4
 
Theorem4bc2eq6 23435 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 4  _C  2 )  =  6
 
Theoremhalfthird 23436 Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  (
 ( 1  /  2
 )  -  ( 1 
 /  3 ) )  =  ( 1  / 
 6 )
 
Theorem5recm6rec 23437 One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 ( 1  /  5
 )  -  ( 1 
 /  6 ) )  =  ( 1  / ; 3 0 )
 
16.7.6  Greatest common divisor and divisibility
 
Theorempdivsq 23438 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( P  e.  Prime  /\  M  e.  ZZ )  ->  ( P  ||  M  <->  P 
 ||  ( M ^
 2 ) ) )
 
Theoremdvdspw 23439 Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  ( K  ||  M  ->  K  ||  ( M ^ N ) ) )
 
Theoremgcd32 23440 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  C )  =  ( ( A 
 gcd  C )  gcd  B ) )
 
Theoremgcdabsorb 23441 Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  B )  =  ( A  gcd  B ) )
 
16.7.7  Properties of relationships
 
Theorembrtp 23442 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  X  e.  _V   &    |-  Y  e.  _V   =>    |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  ( ( X  =  A  /\  Y  =  B )  \/  ( X  =  C  /\  Y  =  D )  \/  ( X  =  E  /\  Y  =  F ) ) )
 
Theoremdftr6 23443 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
 |-  A  e.  _V   =>    |-  ( Tr  A  <->  A  e.  ( _V  \  ran  ( (  _E  o.  _E  )  \  _E  ) ) )
 
Theoremcoep 23444* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A (  _E  o.  R ) B  <->  E. x  e.  B  A R x )
 
Theoremcoepr 23445* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A ( R  o.  `'  _E  ) B  <->  E. x  e.  A  x R B )
 
Theoremdffr5 23446 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
 |-  ( R  Fr  A  <->  ( ~P A  \  { (/) } )  C_  ran  (  _E  \  (  _E  o.  `' R ) ) )
 
Theoremdfso2 23447 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Or  A  <->  ( R  Po  A  /\  ( A  X.  A )  C_  ( R  u.  (  _I  u.  `' R ) ) ) )
 
Theoremdfpo2 23448 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Po  A  <->  ( ( R  i^i  (  _I  |`  A ) )  =  (/)  /\  (
 ( R  i^i  ( A  X.  A ) )  o.  ( R  i^i  ( A  X.  A ) ) )  C_  R ) )
 
Theorembr8 23449* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  (
 g  =  G  ->  ( ze  <->  si ) )   &    |-  ( h  =  H  ->  (
 si 
 <->  rh ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  E. g  e.  P  E. h  e.  P  ( p  = 
 <. <. a ,  b >. ,  <. c ,  d >.
 >.  /\  q  =  <. <.
 e ,  f >. , 
 <. g ,  h >. >.  /\  ph ) }   =>    |-  ( ( ( X  e.  S  /\  A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q  /\  E  e.  Q )  /\  ( F  e.  Q  /\  G  e.  Q  /\  H  e.  Q )
 )  ->  ( <. <. A ,  B >. , 
 <. C ,  D >. >. R <. <. E ,  F >. ,  <. G ,  H >.
 >. 
 <->  rh ) )
 
Theorembr6 23450* Substitution for an six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  ( p  = 
 <. a ,  <. b ,  c >. >.  /\  q  =  <. d ,  <. e ,  f >. >.  /\  ph ) }   =>    |-  (
 ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q  /\  C  e.  Q )  /\  ( D  e.  Q  /\  E  e.  Q  /\  F  e.  Q )
 )  ->  ( <. A ,  <. B ,  C >.
 >. R <. D ,  <. E ,  F >. >.  <->  ze ) )
 
Theorembr4 23451* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( p  = 
 <. a ,  b >.  /\  q  =  <. c ,  d >.  /\  ph ) }   =>    |-  ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q ) )  ->  ( <. A ,  B >. R <. C ,  D >.  <->  ta ) )
 
Theoremdfres3 23452 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A )
 )
 
Theoremcnvco1 23453 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( `' A  o.  B )  =  ( `' B  o.  A )
 
Theoremcnvco2 23454 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( A  o.  `' B )  =  ( B  o.  `' A )
 
16.7.8  Properties of functions and mappings
 
Theoremfunpsstri 23455 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
 |-  (
 ( Fun  H  /\  ( F  C_  H  /\  G  C_  H )  /\  ( dom  F  C_  dom  G  \/  dom  G  C_  dom  F ) )  ->  ( F 
 C.  G  \/  F  =  G  \/  G  C.  F ) )
 
Theoremfundmpss 23456 If a class  F is a proper subset of a function  G, then  dom  F  C.  dom  G. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( Fun  G  ->  ( F  C.  G  ->  dom  F  C.  dom 
 G ) )
 
Theoremfvresval 23457 The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( ( F  |`  B ) `
  A )  =  ( F `  A )  \/  ( ( F  |`  B ) `  A )  =  (/) )
 
Theoremmptrel 23458 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Rel  ( x  e.  A  |->  B )
 
Theoremfunsseq 23459 Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  (
 ( Fun  F  /\  Fun 
 G  /\  dom  F  =  dom  G )  ->  ( F  =  G  <->  F  C_  G ) )
 
Theoremfununiq 23460 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( Fun  F  ->  ( ( A F B  /\  A F C ) 
 ->  B  =  C ) )
 
Theoremfunbreq 23461 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( ( Fun  F  /\  A F B ) 
 ->  ( A F C  <->  B  =  C ) )
 
Theoremmpteq12d 23462 An equality inference for the maps to notation. Compare mpteq12dv 4038. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
 
Theoremfprb 23463* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  =/=  B  ->  ( F : { A ,  B } --> R  <->  E. x  e.  R  E. y  e.  R  F  =  { <. A ,  x >. ,  <. B ,  y >. } ) )
 
Theorembr1steq 23464 Uniqueness condition for binary relationship over the  1st relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >. 1st C  <->  C  =  A )
 
Theorembr2ndeq 23465 Uniqueness condition for binary relationship over the  2nd relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >. 2nd C  <->  C  =  B )
 
Theoremdfdm5 23466 Definition of domain in terms of 
1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  dom  A  =  ( ( 1st  |`  ( _V  X.  _V ) ) " A )
 
Theoremdfrn5 23467 Definition of range in terms of 
2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A )
 
16.7.9  Epsilon induction
 
Theoremsetinds 23468* Principle of  _E induction (set induction). If a property passes from all elements of  x to  x itself, then it holds for all  x. (Contributed by Scott Fenton, 10-Mar-2011.)
 |-  ( A. y  e.  x  [. y  /  x ]. ph 
 ->  ph )   =>    |-  ph
 
Theoremsetinds2f 23469*  _E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ x ps   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   &    |-  ( A. y  e.  x  ps  ->  ph )   =>    |-  ph
 
Theoremsetinds2 23470*  _E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( A. y  e.  x  ps  ->  ph )   =>    |-  ph
 
16.7.10  Ordinal numbers
 
Theoremelpotr 23471* A class of transitive sets is partially ordered by  _E. (Contributed by Scott Fenton, 15-Oct-2010.)
 |-  ( A. z  e.  A  Tr  z  ->  _E  Po  A )
 
Theoremdford5reg 23472 Given ax-reg 7239, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)
 |-  ( Ord  A  <->  ( Tr  A  /\  _E  Or  A ) )
 
Theoremdfon2lem1 23473 Lemma for dfon2 23482. (Contributed by Scott Fenton, 28-Feb-2011.)
 |-  Tr  U.
 { x  |  (
 ph  /\  Tr  x  /\  ps ) }
 
Theoremdfon2lem2 23474* Lemma for dfon2 23482 (Contributed by Scott Fenton, 28-Feb-2011.)
 |-  U. { x  |  ( x  C_  A  /\  ph  /\  ps ) }  C_  A
 
Theoremdfon2lem3 23475* Lemma for dfon2 23482. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  ->  ( Tr  A  /\  A. z  e.  A  -.  z  e.  z ) ) )
 
Theoremdfon2lem4 23476* Lemma for dfon2 23482. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  /\  A. y ( ( y 
 C.  B  /\  Tr  y )  ->  y  e.  B ) )  ->  ( A  C_  B  \/  B  C_  A ) )
 
Theoremdfon2lem5 23477* Lemma for dfon2 23482. Two sets satisfying the new definition also satisfy trichotomy with respect to 
e. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  /\  A. y ( ( y 
 C.  B  /\  Tr  y )  ->  y  e.  B ) )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A )
 )
 
Theoremdfon2lem6 23478* Lemma for dfon2 23482. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  (
 ( Tr  S  /\  A. x  e.  S  A. z ( ( z 
 C.  x  /\  Tr  z )  ->  z  e.  x ) )  ->  A. y ( ( y 
 C.  S  /\  Tr  y )  ->  y  e.  S ) )
 
Theoremdfon2lem7 23479* Lemma for dfon2 23482. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  A  e.  _V   =>    |-  ( A. x ( ( x  C.  A  /\  Tr  x )  ->  x  e.  A )  ->  ( B  e.  A  ->  A. y ( ( y  C.  B  /\  Tr  y )  ->  y  e.  B ) ) )
 
Theoremdfon2lem8 23480* Lemma for dfon2 23482. The intersection of a non-empty class  A of new ordinals is itself a new ordinal and is contained within  A (Contributed by Scott Fenton, 26-Feb-2011.)
 |-  (
 ( A  =/=  (/)  /\  A. x  e.  A  A. y
 ( ( y  C.  x  /\  Tr  y ) 
 ->  y  e.  x ) )  ->  ( A. z ( ( z 
 C.  |^| A  /\  Tr  z )  ->  z  e. 
 |^| A )  /\  |^|
 A  e.  A ) )
 
Theoremdfon2lem9 23481* Lemma for dfon2 23482. A class of new ordinals is well-founded by  _E. (Contributed by Scott Fenton, 3-Mar-2011.)
 |-  ( A. x  e.  A  A. y ( ( y 
 C.  x  /\  Tr  y )  ->  y  e.  x )  ->  _E  Fr  A )
 
Theoremdfon2 23482*  On consists of all sets that contain all its transitive proper subsets. This definition comes from J. R. Isbell, "A Definition of Ordinal Numbers," American Mathematical Monthly, vol 67 (1960), pp. 51-52. (Contributed by Scott Fenton, 20-Feb-2011.)
 |-  On  =  { x  |  A. y ( ( y 
 C.  x  /\  Tr  y )  ->  y  e.  x ) }
 
Theoremdomep 23483 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)
 |-  dom  _E  =  _V
 
Theoremrdgprc0 23484 The value of the recursive definition generator at  (/) when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
 
Theoremrdgprc 23485 The value of the recursive definition generator when  I is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( -.  I  e.  _V  ->  rec ( F ,  I )  =  rec ( F ,  (/) ) )
 
Theoremdfrdg2 23486* Alternate definition of the recursive function generator when  I is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( I  e.  V  ->  rec ( F ,  I
 )  =  U. {
 f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  if ( y  =  (/) ,  I ,  if ( Lim  y ,  U. ( f "
 y ) ,  ( F `  ( f `  U. y ) ) ) ) ) } )
 
Theoremdfrdg3 23487* Generalization of dfrdg2 23486 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  rec ( F ,  I )  =  U. { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  if (
 y  =  (/) ,  if ( I  e.  _V ,  I ,  (/) ) ,  if ( Lim  y ,  U. ( f "
 y ) ,  ( F `  ( f `  U. y ) ) ) ) ) }
 
16.7.11  Defined equality axioms
 
Theoremaxextdfeq 23488 A version of ax-ext 2237 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
 |-  E. z
 ( ( z  e.  x  ->  z  e.  y )  ->  ( ( z  e.  y  ->  z  e.  x )  ->  ( x  e.  w  ->  y  e.  w ) ) )
 
Theoremax13dfeq 23489 A version of ax-13 1625 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
 |-  E. z
 ( ( z  e.  x  ->  z  e.  y )  ->  ( w  e.  x  ->  w  e.  y ) )
 
Theoremaxextdist 23490 ax-ext 2237 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  (
 ( -.  A. z  z  =  x  /\  -. 
 A. z  z  =  y )  ->  ( A. z ( z  e.  x  <->  z  e.  y
 )  ->  x  =  y ) )
 
Theoremaxext4dist 23491 axext4 2240 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  (
 ( -.  A. z  z  =  x  /\  -. 
 A. z  z  =  y )  ->  ( x  =  y  <->  A. z ( z  e.  x  <->  z  e.  y
 ) ) )
 
Theorem19.12b 23492* 19.12vv 2032 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x A. y (
 ph  ->  ps )  <->  A. y E. x ( ph  ->  ps )
 )
 
Theoremexnel 23493 There is always a set not in  y. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  E. x  -.  x  e.  y
 
Theoremdistel 23494 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4130 and elirrv 7244.) (Contributed by Scott Fenton, 15-Dec-2010.)
 |-  ( -.  A. y  y  =  x  <->  -.  A. y  -.  x  e.  y )
 
Theoremaxextndbi 23495 axextnd 8146 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
 |-  E. z
 ( x  =  y  <-> 
 ( z  e.  x  <->  z  e.  y ) )
 
16.7.12  Hypothesis builders
 
Theoremhbntg 23496 A more general form of hbnt 1717. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( A. x ( ph  ->  A. x ps )  ->  ( -.  ps  ->  A. x  -.  ph ) )
 
Theoremhbimtg 23497 A more general and closed form of hbim 1723. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  (
 ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  (
 ( ch  ->  ps )  ->  A. x ( ph  ->  th ) ) )
 
Theoremhbaltg 23498 A more general and closed form of hbal 1567. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( A. x ( ph  ->  A. y ps )  ->  ( A. x ph  ->  A. y A. x ps ) )
 
Theoremhbng 23499 A more general form of hbn 1722. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( ph  ->  A. x ps )   =>    |-  ( -.  ps  ->  A. x  -.  ph )
 
Theoremhbimg 23500 A more general form of hbim 1723. (Contributed by Scott Fenton, 13-Dec-2010.)
 |-  ( ph  ->  A. x ps )   &    |-  ( ch  ->  A. x th )   =>    |-  (
 ( ps  ->  ch )  ->  A. x ( ph  ->  th ) )
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