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Theorem List for Metamath Proof Explorer - 28101-28200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj1309 28101* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   =>    |-  ( w  e.  B  ->  A. x  w  e.  B )
 
Theorembnj1307 28102* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( w  e.  B  ->  A. x  w  e.  B )   =>    |-  ( w  e.  C  ->  A. x  w  e.  C )
 
Theorembnj1311 28103* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   =>    |-  ( ( R 
 FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  (
 g  |`  D )  =  ( h  |`  D ) )
 
Theorembnj1318 28104 Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( X  =  Y  ->  trCl
 ( X ,  A ,  R )  =  trCl ( Y ,  A ,  R ) )
 
Theorembnj1326 28105* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   =>    |-  ( ( R 
 FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  (
 g  |`  D )  =  ( h  |`  D ) )
 
Theorembnj1321 28106* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  E. f ta )  ->  E! f ta )
 
Theorembnj1364 28107 Property of  FrSe. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( R  FrSe  A  ->  R  Se  A )
 
Theorembnj1371 28108* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  ( ta'  <->  ( f  e.  C  /\  dom  f  =  ( {
 y }  u.  trCl ( y ,  A ,  R ) ) ) )   =>    |-  ( f  e.  H  ->  Fun  f )
 
Theorembnj1373 28109* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  ( ta'  <->  [. y  /  x ].
 ta )   =>    |-  ( ta'  <->  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
 
Theorembnj1374 28110* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   =>    |-  ( f  e.  H  ->  f  e.  C )
 
Theorembnj1384 28111* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   =>    |-  ( R  FrSe  A  ->  Fun  P )
 
Theorembnj1388 28112* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   =>    |-  ( ch  ->  A. y  e.  pred  ( x ,  A ,  R ) E. f ta' )
 
Theorembnj1398 28113* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  ( th 
 <->  ( ch  /\  z  e.  U_ y  e.  pred  ( x ,  A ,  R ) ( {
 y }  u.  trCl ( y ,  A ,  R ) ) ) )   &    |-  ( et  <->  ( th  /\  y  e.  pred ( x ,  A ,  R )  /\  z  e.  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   =>    |-  ( ch  ->  U_ y  e.  pred  ( x ,  A ,  R )
 ( { y }  u.  trCl ( y ,  A ,  R ) )  =  dom  P )
 
Theorembnj1413 28114* Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  B  e.  _V )
 
Theorembnj1408 28115* Technical lemma for bnj1414 28116. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   &    |-  C  =  ( 
 pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   =>    |-  ( ( R 
 FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
 
Theorembnj1414 28116* Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
 
Theorembnj1415 28117* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   =>    |-  ( ch  ->  dom  P  =  trCl ( x ,  A ,  R ) )
 
Theorembnj1416 28118 Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  dom 
 P  =  trCl ( x ,  A ,  R ) )   =>    |-  ( ch  ->  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
 
Theorembnj1418 28119 Property of  pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 y  e.  pred ( x ,  A ,  R )  ->  y R x )
 
Theorembnj1417 28120* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  ( ph 
 <->  R  FrSe  A )   &    |-  ( ps 
 <->  -.  x  e.  trCl ( x ,  A ,  R ) )   &    |-  ( ch 
 <-> 
 A. y  e.  A  ( y R x 
 ->  [. y  /  x ].
 ps ) )   &    |-  ( th 
 <->  ( ph  /\  x  e.  A  /\  ch )
 )   &    |-  B  =  (  pred ( x ,  A ,  R )  u.  U_ y  e.  pred  ( x ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ph  ->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
 
Theorembnj1421 28121* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  Fun 
 P )   &    |-  ( ch  ->  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )   &    |-  ( ch  ->  dom  P  =  trCl ( x ,  A ,  R ) )   =>    |-  ( ch  ->  Fun 
 Q )
 
Theorembnj1444 28122* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   =>    |-  ( rh  ->  A. y rh )
 
Theorembnj1445 28123* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   &    |-  ( si  <->  ( rh  /\  y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   &    |-  ( ph  <->  ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  (
 f `  x )  =  ( G `  Y ) ) )   &    |-  X  =  <. z ,  (
 f  |`  pred ( z ,  A ,  R ) ) >.   =>    |-  ( si  ->  A. d si )
 
Theorembnj1446 28124* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. d ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1447 28125* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. y ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1448 28126* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. f ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1449 28127* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   =>    |-  ( ze  ->  A. f ze )
 
Theorembnj1442 28128* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   =>    |-  ( et  ->  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1450 28129* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   &    |-  ( si  <->  ( rh  /\  y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   &    |-  ( ph  <->  ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  (
 f `  x )  =  ( G `  Y ) ) )   &    |-  X  =  <. z ,  (
 f  |`  pred ( z ,  A ,  R ) ) >.   =>    |-  ( ze  ->  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1423 28130* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   =>    |-  ( ch  ->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1452 28131* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   =>    |-  ( ch  ->  E  e.  B )
 
Theorembnj1466 28132* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( w  e.  Q  ->  A. f  w  e.  Q )
 
Theorembnj1467 28133* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( w  e.  Q  ->  A. d  w  e.  Q )
 
Theorembnj1463 28134* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  e.  _V )   &    |-  ( ch  ->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )   &    |-  ( ch  ->  Q  Fn  E )   &    |-  ( ch  ->  E  e.  B )   =>    |-  ( ch  ->  Q  e.  C )
 
Theorembnj1489 28135* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( ch  ->  Q  e.  _V )
 
Theorembnj1491 28136* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  ( Q  e.  C  /\  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )   =>    |-  ( ( ch  /\  Q  e.  _V )  ->  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
 
Theorembnj1312 28137* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
 
Theorembnj1493 28138* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )
 
Theorembnj1497 28139* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  A. g  e.  C  Fun  g
 
Theorembnj1498 28140* Technical lemma for bnj60 28141. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  dom 
 F  =  A )
 
18.24.5  Well-founded recursion, part 1 of 3
 
Theorembnj60 28141* Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  F  Fn  A )
 
Theorembnj1514 28142* Technical lemma for bnj1500 28147. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  (
 f  e.  C  ->  A. x  e.  dom  f
 ( f `  x )  =  ( G `  Y ) )
 
Theorembnj1518 28143* Technical lemma for bnj1500 28147. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  x  e.  A ) )   &    |-  ( ps  <->  ( ph  /\  f  e.  C  /\  x  e. 
 dom  f ) )   =>    |-  ( ps  ->  A. d ps )
 
Theorembnj1519 28144* Technical lemma for bnj1500 28147. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. )  ->  A. d
 ( F `  x )  =  ( G ` 
 <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1520 28145* Technical lemma for bnj1500 28147. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. )  ->  A. f
 ( F `  x )  =  ( G ` 
 <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1501 28146* Technical lemma for bnj1500 28147. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  x  e.  A ) )   &    |-  ( ps  <->  ( ph  /\  f  e.  C  /\  x  e. 
 dom  f ) )   &    |-  ( ch  <->  ( ps  /\  d  e.  B  /\  dom  f  =  d ) )   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
18.24.6  Well-founded recursion, part 2 of 3
 
Theorembnj1500 28147* Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1525 28148* Technical lemma for bnj1522 28151. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) ) )   &    |-  ( ps  <->  ( ph  /\  F  =/=  H ) )   =>    |-  ( ps  ->  A. x ps )
 
Theorembnj1529 28149* Technical lemma for bnj1522 28151. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )   &    |-  ( w  e.  F  ->  A. x  w  e.  F )   =>    |-  ( ch  ->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
 
Theorembnj1523 28150* Technical lemma for bnj1522 28151. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) ) )   &    |-  ( ps  <->  ( ph  /\  F  =/=  H ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  A  /\  ( F `
  x )  =/=  ( H `  x ) ) )   &    |-  D  =  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }   &    |-  ( th 
 <->  ( ch  /\  y  e.  D  /\  A. z  e.  D  -.  z R y ) )   =>    |-  ( ( R 
 FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) )  ->  F  =  H )
 
18.24.7  Well-founded recursion, part 3 of 3
 
Theorembnj1522 28151* Well-founded recursion, part 3 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) )  ->  F  =  H )
 
18.25  Mathbox for Norm Megill

Note: A label suffixed with "N" (after the "Atoms..." section below), such as lshpnel2N 28342, means that the definition or theorem is not used for the derivation of hlathil 31321. This is a temporary renaming to assist cleaning up the theorems needed by hlathil 31321.

Please inform me of any changes that might affect my mathbox, since I may be working on it independently of the github commits. - Norm 30-Nov-2015

 
18.25.1  Study of ax-6, ax-7, ax-11, ax-12
 
TheoremequidK 28152 (Theorems equidK 28152 through ax12dgen4K 28197 are part of a study of our non-Tarski predicate calculus axiom schemes. We are using this theorem as a placeholder to describe this study.)

The orginal axiom schemes of Tarski's predicate calculus are ax-5 1533, ax-8 1623, ax-9v 1632, ax-13 1625, ax-14 1626, and ax-17 1628 (see http://us.metamath.org/mpegif/mmset.html#compare) and are shown as axiom schemes B4 through B8 in [KalishMontague] p. 81. These are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85.

The axiom system of set.mm includes the additional axiom schemes ax-6 1534, ax-7 1535, ax-11 1624, and ax-12 1633, which are not part of Tarski's axiom schemes. They are used (and we conjecture are required) to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and set metavariables, bundled or not, whose object-language instances are valid. (ax-11 1624 has been proved to be required; see http://us.metamath.org/award2003.html#9a. Metalogical independence of the other three are open problems.)

(There are additional predicate calculus axiom schemes included in set.mm such as ax-4 1692, but they can all be proved as theorems from the above.)

Terminology: Two set (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the  x and  y in ax-9 1684 are bundled, but they are not in ax-9v 1632. We also say that a scheme is bundled when it has at least one pair of bundled set metavariables. If distinct variable conditions are added to all set metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax-9v 1632 is the principal instance of ax-9 1684. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance  -.  A. x -.  x  =  x of ax-9 1684 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($d) to be concerned with. There is also a small economy in being able to state principal and degenerate instances simultaneously. A disadvantage is that bundling may present difficulties in translations to other proof languages, which typically lack the concept (in part because their variables often represent the variables of the object language rather than metavariables ranging over them).

Because Tarski's axiom schemes are logically complete, they can be used to prove any object-language instance of ax-6 1534, ax-7 1535, ax-11 1624, and ax-12 1633. "Translating" this to Metamath, it means that Tarksi's axioms can prove any substitution instance of ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633 in which (1) there are no wff metavariables and (2) all set metavariables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each set metavariable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.)

It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes.

Past work showed that instances of ax-11o 1941 meeting condition (1) can be proved without invoking that axiom scheme (see comments in ax-11 1624). However, it was somewhat awkward to use, involving an inductive argument with auxiliary theorems ax11eq 2109, ax11el 2110, ax11indn 2112, ax11indi 2113, and ax11inda 2117. It also used axiom schemes other than Tarski's.

The new theorem schemes ax6wK 28182, ax7wK 28185, ax11wK 28190, and ax12wK 28193 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-6 1534, ax-7 1535, ax-11 1624, and ax-12 1633 meeting conditions (1) and (2). (The "K" suffix stands for Kalish/Montague, whose paper was a source for some of the proofs. I may change these names in the future since our practice has been to reserve upper case for special cases such as the ALT or OLD suffixes.) Each hypothesis of ax6wK 28182, ax7wK 28185, and ax11wK 28190 is of the form  ( x  =  y  ->  ( ph  <->  ps ) ) where  ps is an auxiliary or "dummy" wff metavariable in which  x doesn't occur. We can show by induction on formula length that the hypotheses can be eliminated in all cases meeting conditions (1) and (2). The example ax11wdemoK 28192 illustrates the techniques (equality theorems and bound variable renaming) used to achieve this.

We also show the degenerate instances for axioms with bundled variables in ax7dgenK 28187, ax11dgenK 28191, ax12dgen1K 28194, ax12dgen2K 28195, ax12dgen3K 28196, and ax12dgen4K 28197. (Their proofs are trivial but we include them to be thorough.) Combining the principal and degenerate cases outside of Metamath, we show that the bundled schemes ax-6 1534, ax-7 1535, ax-11 1624, and ax-12 1633 are schemes of Tarski's system, meaning that all object language instances they generate are theorems of Tarski's system.

It is interesting that Tarski's system bundles set metavariables in ax-8 1623, ax-13 1625, and ax-14 1626; indeed, a degenerate instance of ax-8 1623 appears to be indispensable for the proof of equidK 28152. Perhaps his general philosophy was that bundling is acceptable for free variables. But he also used the bundled scheme ax-9 1684 in an older system, so it seems the main purpose of his later ax-9v 1632 was just to show that the weaker unbundled form is sufficient rather than an aesthetic objection to bundled free and bound variables.

The case of ax-4 1692 is curious: originally an axiom of Tarski's system, it was proved redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the compact scheme form  A. x ph  ->  ph apparently cannot be proved directly from Tarski's other axioms. The best we can do seems to be ax4wK 28169, again requiring substitution instances of  ph that meet conditions (1) and (2) above. Note that our direct proof ax4 1691 requires ax-11 1624, which is not part of Tarski's system.

(End of study description.)

Identity law for equality. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. Lemma 2 of [KalishMontague] p. 85. (Contributed by NM, 9-Apr-2017.)

 |-  x  =  x
 
TheoremequcomiK 28153 Commutative law for equality. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). Lemma 3 of [KalishMontague] p. 85. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  y  =  x )
 
Theoremequequ1K 28154 An equality law for equality. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  ( x  =  z  <->  y  =  z
 ) )
 
Theoremequequ2K 28155 An equality law for equality. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  ( z  =  x  <->  z  =  y
 ) )
 
Theoremelequ1K 28156 An equality law for the membership predicate. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  ( x  e.  z  <->  y  e.  z
 ) )
 
Theoremelequ2K 28157 An equality for the membership predicate. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  ( z  e.  x  <->  z  e.  y
 ) )
 
TheoremalimiK 28158 Add universal quantifier to both sides of an implication. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 9-Apr-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
TheoremalbiiK 28159 Add universal quantifier to both sides of an equivalence. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). Part of Lemma 5 of [KalishMontague] p. 86. (The other parts are just notbii 289 and imbi12i 318.) (Contributed by NM, 9-Apr-2017.)
 |-  ( ph 
 <->  ps )   =>    |-  ( A. x ph  <->  A. x ps )
 
TheoremalimdK 28160 Add universal quantifier to both sides of an equivalence. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). Part of Lemma 5 of [KalishMontague] p. 86. (The other parts are just notbii 289 and imbi12i 318.) (Contributed by NM, 19-Apr-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
 
TheoremalimdvK 28161* Add universal quantifier to both sides of an equivalence. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). Part of Lemma 5 of [KalishMontague] p. 86. (The other parts are just notbii 289 and imbi12i 318.) (Contributed by NM, 11-Apr-2017.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
 
TheoremalbidK 28162 Add universal quantifier to both sides of an equivalence. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 19-Apr-2017.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
TheoremalbidvK 28163* Add universal quantifier to both sides of an equivalence. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 11-Apr-2017.)
 |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theorema4eimfK 28164 Specialization, with additional weakening to allow bundling of  x and  y. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 23-Apr-1017.)
 |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( -.  A. x  -.  x  =  y 
 ->  ( A. x ph  ->  E. x ps )
 )
 
Theorema4imfK 28165 Specialization, with additional weakening to allow bundling of  x and  y. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 23-Apr-1017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( -.  A. x  -.  x  =  y 
 ->  ( A. x ph  ->  ps ) )
 
Theorema4imK 28166* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 19-Apr-2017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theorema4imvK 28167* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremax4wfK 28168* Weak version of ax-4 1692. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). Lemma 9 of [KalishMontague] p. 86. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.)
 |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. x ph 
 ->  A. y A. x ph )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  ph )
 
Theoremax4wK 28169* Weak version of ax-4 1692. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). Lemma 9 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  ph )
 
Theoremax4vK 28170* Version of ax-4 1692 when  x does not occur in  ph. This provides the other direction of ax-17 1628. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 10-Apr-2017.)
 |-  ( A. x ph  ->  ph )
 
Theorem19.8vK 28171* Version of 19.8a 1758 and its converse when  x does not occur in  ph. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 10-Apr-2017.)
 |-  ( ph 
 <-> 
 E. x ph )
 
Theoremax4truK 28172 Version of ax-4 1692 when  ph is true. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 23-Apr-1017.)
 |-  ph   =>    |-  ( A. x ph  -> 
 ph )
 
Theoremax4falK 28173 Version of ax-4 1692 when  ph is false. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 23-Apr-1017.)
 |-  -.  ph   =>    |-  ( A. x ph  ->  ph )
 
Theoremax9dgenK 28174 Degenerate case of ax-9 1684. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 23-Apr-2017.)
 |-  -.  A. x  -.  x  =  x
 
Theorem19.2K 28175 Theorem 19.2 of [Margaris] p. 89. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 23-Apr-1017.)
 |-  ( A. x ph  ->  E. x ph )
 
TheoremcbvaliK 28176* Change bound variable. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
TheoremcbvalivK 28177* Change bound variable. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  A. y ps )
 
TheoremcbvalK 28178* Change bound variable. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 9-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. y ps  ->  A. x A. y ps )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
TheoremcbvalvK 28179* Change bound variable. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
TheoremcbvexvK 28180* Change bound variable. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
Theoremax6wfK 28181* Weak version of ax-6 1534 from which we can prove any ax-6 1534 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 19-Apr-2017.)
 |-  ( A. x ph  ->  A. y A. x ph )   &    |-  ( -.  ps  ->  A. x  -.  ps )   &    |-  ( A. y ps  ->  A. x A. y ps )   &    |-  ( -.  ph  ->  A. y  -.  ph )   &    |-  ( -.  A. y ps  ->  A. x  -.  A. y ps )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x ph 
 ->  A. x  -.  A. x ph )
 
Theoremax6wK 28182* Weak version of ax-6 1534 from which we can prove any ax-6 1534 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( -.  A. x ph  ->  A. x  -.  A. x ph )
 
Theoremhba1wK 28183* Weak version of hba1 1718. See comments for ax6wK 28182. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 9-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ph 
 ->  A. x A. x ph )
 
Theoremhbe1wK 28184* Weak version of hbe1 1565. See comments for ax6wK 28182. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ph 
 ->  A. x E. x ph )
 
Theoremax7wK 28185* Weak version of ax-7 1535 from which we can prove any ax-7 1535 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). Unlike ax-7 1535, this theorem requires that  x and  y be distinct i.e. are not bundled. See the description in the comment of equidK 28152. (Contributed by NM, 10-Apr-2017.)
 |-  (
 y  =  z  ->  ( ph  <->  ps ) )   =>    |-  ( A. x A. y ph  ->  A. y A. x ph )
 
TheoremhbalwK 28186* Weak version of hbal 1567. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). Unlike hbal 1567, this theorem requires that  x and  y be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.)
 |-  ( x  =  z  ->  (
 ph 
 <->  ps ) )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( A. y ph  ->  A. x A. y ph )
 
Theoremax7dgenK 28187 Degenerate instance of ax-7 1535 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 13-Apr-2017.)
 |-  ( A. x A. x ph  ->  A. x A. x ph )
 
Theoremax11wflemK 28188 Verson of ax11wlemK 28189 without distinct variables. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 19-Apr-2017.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax11wlemK 28189* Lemma for weak version of ax-11 1624. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). In some cases, this lemma may lead to shorter proofs than ax11wK 28190. (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremax11wK 28190* Weak version of ax-11 1624 from which we can prove any ax-7 1535 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). An instance of the first hypothesis will normally require that  x and  y be distinct (unless  x does not occur in  ph). See the description in the comment of equidK 28152. (Contributed by NM, 10-Apr-2017.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  z  ->  ( ph  <->  ch ) )   =>    |-  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremax11dgenK 28191 Degenerate instance of ax-11 1624 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 13-Apr-2017.)
 |-  ( x  =  x  ->  (
 A. x ph  ->  A. x ( x  =  x  ->  ph ) ) )
 
Theoremax11wdemoK 28192* Example of an application of ax11wK 28190 that results in an instance of ax-11 1624 for a contrived formula with mixed free and bound variables,  ( x  e.  y  /\  A. x
z  e.  x  /\  A. y A. z y  e.  x ), in place of  ph. The proof illustrates bound variable renaming with cbvalvK 28179 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 14-Apr-2017.)
 |-  ( x  =  y  ->  (
 A. y ( x  e.  y  /\  A. x  z  e.  x  /\  A. y A. z  y  e.  x )  ->  A. x ( x  =  y  ->  ( x  e.  y  /\  A. x  z  e.  x  /\  A. y A. z  y  e.  x )
 ) ) )
 
Theoremax12wK 28193* Weak version (principal instance) of ax-12 1633 not involving bundling. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). The proof is trivial but is included to complete the set ax6wK 28182, ax7wK 28185, and ax11wK 28190. See the description in the comment of equidK 28152. (Contributed by NM, 10-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  z 
 ->  A. x  y  =  z ) )
 
Theoremax12dgen1K 28194 Degenerate instance of ax-12 1633 where bundled variables  x and  y have a common substitution. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  x  ->  ( x  =  z 
 ->  A. x  x  =  z ) )
 
Theoremax12dgen2K 28195 Degenerate instance of ax-12 1633 where bundled variables  x and  z have a common substitution. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  x 
 ->  A. x  y  =  x ) )
 
Theoremax12dgen3K 28196 Degenerate instance of ax-12 1633 where bundled variables  y and  z have a common substitution. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  y  ->  ( y  =  y 
 ->  A. x  y  =  y ) )
 
Theoremax12dgen4K 28197 Degenerate instance of ax-12 1633 where bundled variables  x,  y, and  z have a common substitution. Uses only Tarski's FOL axiom schemes (see description for equidK 28152). (Contributed by NM, 13-Apr-2017.)
 |-  ( -.  x  =  x  ->  ( x  =  x 
 ->  A. x  x  =  x ) )
 
18.25.2  Derive ax-12o from ax-12
 
Theoremax12vX 28198* A weaker version of ax-12 1633 with distinct variable restrictions on pairs  x ,  z and  y ,  z. In order to show that this weakening is adequate, this should be the only theorem referencing ax-12 1633 directly. (Contributed by NM, 30-Jun-2016.)
 |-  ( -.  x  =  y  ->  ( y  =  z 
 ->  A. x  y  =  z ) )
 
Theoremax12o10lem1X 28199 Lemma for ax12o 1663 and ax10 1677. Same as equcomi 1822, using only Tarski's FOL axiom schemes (see description for equidK 28152).

Note that in these lemmas we use ax-9v 1632 instead of ax-9 1684 since the proof of ax9 1683 from ax-9v 1632 makes use of ax-12o 1664. The first use of ax-12o 1664 occurs in ax10lem24 1673. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)

 |-  ( x  =  y  ->  y  =  x )
 
Theoremax12o10lem2X 28200 Lemma for ax12o 1663 and ax10 1677. Same as equequ1 1829, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
 |-  ( x  =  y  ->  ( x  =  z  <->  y  =  z
 ) )
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