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Theorem bnj1053 29787
Description: Technical lemma for bnj69 29821. 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 29581 . . . . . 6  |-  ( n  e.  D  ->  n  e.  om )
12 nnord 6712 . . . . . 6  |-  ( n  e.  om  ->  Ord  n )
13 ordfr 5455 . . . . . 6  |-  ( Ord  n  ->  _E  Fr  n )
1411, 12, 133syl 18 . . . . 5  |-  ( n  e.  D  ->  _E  Fr  n )
153, 14bnj769 29575 . . . 4  |-  ( ch 
->  _E  Fr  n )
1615bnj707 29567 . . 3  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  _E  Fr  n )
17 bnj1053.37 . . 3  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i  e.  n  ( rh  ->  et ) )
1816, 17jca 535 . 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 29786 1  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   {cab 2408   A.wral 2776   E.wrex 2777   _Vcvv 3082   [.wsbc 3300    \ cdif 3434    C_ wss 3437   (/)c0 3762   {csn 3997   U_ciun 4297   class class class wbr 4421    _E cep 4760    Fr wfr 4807   Ord word 5439   suc csuc 5442    Fn wfn 5594   ` cfv 5599   omcom 6704    /\ w-bnj17 29493    predc-bnj14 29495    FrSe w-bnj15 29499    trClc-bnj18 29501    TrFow-bnj19 29503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-tr 4517  df-eprel 4762  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-fn 5602  df-om 6705  df-bnj17 29494  df-bnj18 29502
This theorem is referenced by: (None)
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