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Theorem bnj996 28987
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
bnj996.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj996.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj996.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj996.4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
)  /\  z  e.  pred ( y ,  A ,  R ) ) )
bnj996.5  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
bnj996.6  |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i ) ) )
bnj996.13  |-  D  =  ( om  \  { (/)
} )
bnj996.14  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
Assertion
Ref Expression
bnj996  |-  E. f E. n E. i E. m E. p ( th  ->  ( ch  /\ 
ta  /\  et )
)
Distinct variable groups:    A, f,
i, n, y    D, i    R, f, i, n, y    f, X, i, n, y    ch, m, p    et, m, p    th, f,
i, n    ph, i    m, n, th, p
Allowed substitution hints:    ph( y, z, f, m, n, p)    ps( y, z, f, i, m, n, p)    ch( y, z, f, i, n)    th( y, z)    ta( y,
z, f, i, m, n, p)    et( y,
z, f, i, n)    A( z, m, p)    B( y, z, f, i, m, n, p)    D( y,
z, f, m, n, p)    R( z, m, p)    X( z, m, p)

Proof of Theorem bnj996
StepHypRef Expression
1 bnj996.4 . . . . 5  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
)  /\  z  e.  pred ( y ,  A ,  R ) ) )
2 bnj996.1 . . . . . 6  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
3 bnj996.2 . . . . . 6  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
4 bnj996.13 . . . . . 6  |-  D  =  ( om  \  { (/)
} )
5 bnj996.14 . . . . . 6  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
6 bnj996.3 . . . . . 6  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
72, 3, 4, 5, 6bnj917 28966 . . . . 5  |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) ) )
81, 7bnj771 28794 . . . 4  |-  ( th 
->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) ) )
9 3anass 938 . . . . . 6  |-  ( ( ch  /\  i  e.  n  /\  y  e.  ( f `  i
) )  <->  ( ch  /\  ( i  e.  n  /\  y  e.  (
f `  i )
) ) )
10 bnj996.6 . . . . . . 7  |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i ) ) )
1110anbi2i 675 . . . . . 6  |-  ( ( ch  /\  et )  <-> 
( ch  /\  (
i  e.  n  /\  y  e.  ( f `  i ) ) ) )
129, 11bitr4i 243 . . . . 5  |-  ( ( ch  /\  i  e.  n  /\  y  e.  ( f `  i
) )  <->  ( ch  /\  et ) )
13123exbii 1571 . . . 4  |-  ( E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) )  <->  E. f E. n E. i ( ch  /\  et ) )
148, 13sylib 188 . . 3  |-  ( th 
->  E. f E. n E. i ( ch  /\  et ) )
15 bnj996.5 . . . . . . . . . 10  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
166, 4, 15bnj986 28986 . . . . . . . . 9  |-  ( ch 
->  E. m E. p ta )
1716ancli 534 . . . . . . . 8  |-  ( ch 
->  ( ch  /\  E. m E. p ta )
)
18 19.42vv 1848 . . . . . . . 8  |-  ( E. m E. p ( ch  /\  ta )  <->  ( ch  /\  E. m E. p ta ) )
1917, 18sylibr 203 . . . . . . 7  |-  ( ch 
->  E. m E. p
( ch  /\  ta ) )
2019anim1i 551 . . . . . 6  |-  ( ( ch  /\  et )  ->  ( E. m E. p ( ch  /\  ta )  /\  et ) )
21 19.41vv 1843 . . . . . 6  |-  ( E. m E. p ( ( ch  /\  ta )  /\  et )  <->  ( E. m E. p ( ch 
/\  ta )  /\  et ) )
2220, 21sylibr 203 . . . . 5  |-  ( ( ch  /\  et )  ->  E. m E. p
( ( ch  /\  ta )  /\  et ) )
23 df-3an 936 . . . . . 6  |-  ( ( ch  /\  ta  /\  et )  <->  ( ( ch 
/\  ta )  /\  et ) )
24232exbii 1570 . . . . 5  |-  ( E. m E. p ( ch  /\  ta  /\  et )  <->  E. m E. p
( ( ch  /\  ta )  /\  et ) )
2522, 24sylibr 203 . . . 4  |-  ( ( ch  /\  et )  ->  E. m E. p
( ch  /\  ta  /\  et ) )
26252eximi 1564 . . 3  |-  ( E. n E. i ( ch  /\  et )  ->  E. n E. i E. m E. p ( ch  /\  ta  /\  et ) )
2714, 26bnj593 28774 . 2  |-  ( th 
->  E. f E. n E. i E. m E. p ( ch  /\  ta  /\  et ) )
28 19.37v 1840 . . . . . . . . . 10  |-  ( E. p ( th  ->  ( ch  /\  ta  /\  et ) )  <->  ( th  ->  E. p ( ch 
/\  ta  /\  et ) ) )
2928exbii 1569 . . . . . . . . 9  |-  ( E. m E. p ( th  ->  ( ch  /\ 
ta  /\  et )
)  <->  E. m ( th 
->  E. p ( ch 
/\  ta  /\  et ) ) )
3029bnj132 28752 . . . . . . . 8  |-  ( E. m E. p ( th  ->  ( ch  /\ 
ta  /\  et )
)  <->  ( th  ->  E. m E. p ( ch  /\  ta  /\  et ) ) )
3130exbii 1569 . . . . . . 7  |-  ( E. i E. m E. p ( th  ->  ( ch  /\  ta  /\  et ) )  <->  E. i
( th  ->  E. m E. p ( ch  /\  ta  /\  et ) ) )
3231bnj132 28752 . . . . . 6  |-  ( E. i E. m E. p ( th  ->  ( ch  /\  ta  /\  et ) )  <->  ( th  ->  E. i E. m E. p ( ch  /\  ta  /\  et ) ) )
3332exbii 1569 . . . . 5  |-  ( E. n E. i E. m E. p ( th  ->  ( ch  /\ 
ta  /\  et )
)  <->  E. n ( th 
->  E. i E. m E. p ( ch  /\  ta  /\  et ) ) )
3433bnj132 28752 . . . 4  |-  ( E. n E. i E. m E. p ( th  ->  ( ch  /\ 
ta  /\  et )
)  <->  ( th  ->  E. n E. i E. m E. p ( ch  /\  ta  /\  et ) ) )
3534exbii 1569 . . 3  |-  ( E. f E. n E. i E. m E. p
( th  ->  ( ch  /\  ta  /\  et ) )  <->  E. f
( th  ->  E. n E. i E. m E. p ( ch  /\  ta  /\  et ) ) )
3635bnj132 28752 . 2  |-  ( E. f E. n E. i E. m E. p
( th  ->  ( ch  /\  ta  /\  et ) )  <->  ( th  ->  E. f E. n E. i E. m E. p ( ch  /\  ta  /\  et ) ) )
3727, 36mpbir 200 1  |-  E. f E. n E. i E. m E. p ( th  ->  ( ch  /\ 
ta  /\  et )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    \ cdif 3149   (/)c0 3455   {csn 3640   U_ciun 3905   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719
This theorem is referenced by:  bnj1021  28996
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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  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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