Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1053 Structured version   Unicode version

Theorem bnj1053 29347
Description: Technical lemma for bnj69 29381. 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.)
Hypotheses
Ref Expression
bnj1053.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1053.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1053.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1053.4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1053.5  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
bnj1053.6  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
bnj1053.7  |-  D  =  ( om  \  { (/)
} )
bnj1053.8  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj1053.9  |-  ( et  <->  ( ( th  /\  ta  /\ 
ch  /\  ze )  ->  z  e.  B ) )
bnj1053.10  |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. et ) )
bnj1053.37  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i  e.  n  ( rh  ->  et ) )
Assertion
Ref Expression
bnj1053  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Distinct variable groups:    A, f,
i, n, y    z, A, f, i, n    B, f, i, n, z    D, i    R, f, i, n, y    z, R    f, X, i, n, y    z, X    et, j    ta, f,
i, n, z    th, f, i, n, z    i,
j, n    ph, i
Allowed substitution hints:    ph( y, z, f, j, n)    ps( y, z, f, i, j, n)    ch( y, z, f, i, j, n)    th( y,
j)    ta( y, j)    et( y, z, f, i, n)    ze( y, z, f, i, j, n)    rh( y,
z, f, i, j, n)    A( j)    B( y, j)    D( y, z, f, j, n)    R( j)    K( y, z, f, i, j, n)    X( j)

Proof of Theorem bnj1053
StepHypRef Expression
1 bnj1053.1 . 2  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj1053.2 . 2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj1053.3 . 2  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
4 bnj1053.4 . 2  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
5 bnj1053.5 . 2  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
6 bnj1053.6 . 2  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
7 bnj1053.7 . 2  |-  D  =  ( om  \  { (/)
} )
8 bnj1053.8 . 2  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
9 bnj1053.9 . 2  |-  ( et  <->  ( ( th  /\  ta  /\ 
ch  /\  ze )  ->  z  e.  B ) )
10 bnj1053.10 . 2  |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. et ) )
117bnj923 29139 . . . . . 6  |-  ( n  e.  D  ->  n  e.  om )
12 nnord 4855 . . . . . 6  |-  ( n  e.  om  ->  Ord  n )
13 ordfr 4598 . . . . . 6  |-  ( Ord  n  ->  _E  Fr  n )
1411, 12, 133syl 19 . . . . 5  |-  ( n  e.  D  ->  _E  Fr  n )
153, 14bnj769 29133 . . . 4  |-  ( ch 
->  _E  Fr  n )
1615bnj707 29125 . . 3  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  _E  Fr  n )
17 bnj1053.37 . . 3  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i  e.  n  ( rh  ->  et ) )
1816, 17jca 520 . 2  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
(  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et ) ) )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18bnj1052 29346 1  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   _Vcvv 2958   [.wsbc 3163    \ cdif 3319    C_ wss 3322   (/)c0 3630   {csn 3816   U_ciun 4095   class class class wbr 4214    _E cep 4494    Fr wfr 4540   Ord word 4582   suc csuc 4585   omcom 4847    Fn wfn 5451   ` cfv 5456    /\ w-bnj17 29052    predc-bnj14 29054    FrSe w-bnj15 29058    trClc-bnj18 29060    TrFow-bnj19 29062
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-fn 5459  df-bnj17 29053  df-bnj18 29061
  Copyright terms: Public domain W3C validator