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Theorem bnj1053 29006
Description: Technical lemma for bnj69 29040. 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 28798 . . . . . 6  |-  ( n  e.  D  ->  n  e.  om )
12 nnord 4664 . . . . . 6  |-  ( n  e.  om  ->  Ord  n )
13 ordfr 4407 . . . . . 6  |-  ( Ord  n  ->  _E  Fr  n )
1411, 12, 133syl 18 . . . . 5  |-  ( n  e.  D  ->  _E  Fr  n )
153, 14bnj769 28792 . . . 4  |-  ( ch 
->  _E  Fr  n )
1615bnj707 28784 . . 3  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  _E  Fr  n )
17 bnj1053.37 . . 3  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i  e.  n  ( rh  ->  et ) )
1816, 17jca 518 . 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 29005 1  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788   [.wsbc 2991    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   U_ciun 3905   class class class wbr 4023    _E cep 4303    Fr wfr 4349   Ord word 4391   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719    TrFow-bnj19 28721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-fn 5258  df-bnj17 28712  df-bnj18 28720
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