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Theorem bnj996 29815
Description: Technical lemma for bnj69 29868. 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 29794 . . . . 5  |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) ) )
81, 7bnj771 29624 . . . 4  |-  ( th 
->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i ) ) )
9 3anass 995 . . . . . 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 705 . . . . . 6  |-  ( ( ch  /\  et )  <-> 
( ch  /\  (
i  e.  n  /\  y  e.  ( f `  i ) ) ) )
129, 11bitr4i 260 . . . . 5  |-  ( ( ch  /\  i  e.  n  /\  y  e.  ( f `  i
) )  <->  ( ch  /\  et ) )
13123exbii 1731 . . . 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 201 . . 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 29814 . . . . . . . . 9  |-  ( ch 
->  E. m E. p ta )
1716ancli 558 . . . . . . . 8  |-  ( ch 
->  ( ch  /\  E. m E. p ta )
)
18 19.42vv 1847 . . . . . . . 8  |-  ( E. m E. p ( ch  /\  ta )  <->  ( ch  /\  E. m E. p ta ) )
1917, 18sylibr 217 . . . . . . 7  |-  ( ch 
->  E. m E. p
( ch  /\  ta ) )
2019anim1i 576 . . . . . 6  |-  ( ( ch  /\  et )  ->  ( E. m E. p ( ch  /\  ta )  /\  et ) )
21 19.41vv 1842 . . . . . 6  |-  ( E. m E. p ( ( ch  /\  ta )  /\  et )  <->  ( E. m E. p ( ch 
/\  ta )  /\  et ) )
2220, 21sylibr 217 . . . . 5  |-  ( ( ch  /\  et )  ->  E. m E. p
( ( ch  /\  ta )  /\  et ) )
23 df-3an 993 . . . . . 6  |-  ( ( ch  /\  ta  /\  et )  <->  ( ( ch 
/\  ta )  /\  et ) )
24232exbii 1730 . . . . 5  |-  ( E. m E. p ( ch  /\  ta  /\  et )  <->  E. m E. p
( ( ch  /\  ta )  /\  et ) )
2522, 24sylibr 217 . . . 4  |-  ( ( ch  /\  et )  ->  E. m E. p
( ch  /\  ta  /\  et ) )
26252eximi 1719 . . 3  |-  ( E. n E. i ( ch  /\  et )  ->  E. n E. i E. m E. p ( ch  /\  ta  /\  et ) )
2714, 26bnj593 29604 . 2  |-  ( th 
->  E. f E. n E. i E. m E. p ( ch  /\  ta  /\  et ) )
28 19.37v 1837 . . . . . . . . . 10  |-  ( E. p ( th  ->  ( ch  /\  ta  /\  et ) )  <->  ( th  ->  E. p ( ch 
/\  ta  /\  et ) ) )
2928exbii 1729 . . . . . . . . 9  |-  ( E. m E. p ( th  ->  ( ch  /\ 
ta  /\  et )
)  <->  E. m ( th 
->  E. p ( ch 
/\  ta  /\  et ) ) )
3029bnj132 29581 . . . . . . . 8  |-  ( E. m E. p ( th  ->  ( ch  /\ 
ta  /\  et )
)  <->  ( th  ->  E. m E. p ( ch  /\  ta  /\  et ) ) )
3130exbii 1729 . . . . . . 7  |-  ( E. i E. m E. p ( th  ->  ( ch  /\  ta  /\  et ) )  <->  E. i
( th  ->  E. m E. p ( ch  /\  ta  /\  et ) ) )
3231bnj132 29581 . . . . . 6  |-  ( E. i E. m E. p ( th  ->  ( ch  /\  ta  /\  et ) )  <->  ( th  ->  E. i E. m E. p ( ch  /\  ta  /\  et ) ) )
3332exbii 1729 . . . . 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 29581 . . . 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 1729 . . 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 29581 . 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 214 1  |-  E. f E. n E. i E. m E. p ( th  ->  ( ch  /\ 
ta  /\  et )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898   {cab 2448   A.wral 2749   E.wrex 2750    \ cdif 3413   (/)c0 3743   {csn 3980   U_ciun 4292   suc csuc 5444    Fn wfn 5596   ` cfv 5601   omcom 6719    /\ w-bnj17 29540    predc-bnj14 29542    FrSe w-bnj15 29546    trClc-bnj18 29548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-tr 4512  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-fn 5604  df-om 6720  df-bnj17 29541  df-bnj18 29549
This theorem is referenced by:  bnj1021  29824
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