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Theorem bnj1033 33122
Description: Technical lemma for bnj69 33163. 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
bnj1033.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1033.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1033.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1033.4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1033.5  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
bnj1033.6  |-  ( et  <->  z  e.  trCl ( X ,  A ,  R )
)
bnj1033.7  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
bnj1033.8  |-  D  =  ( om  \  { (/)
} )
bnj1033.9  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj1033.10  |-  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)
Assertion
Ref Expression
bnj1033  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Distinct variable groups:    A, f,
i, n, y    z, A, f, i, n    z, B    D, i    R, f, i, n, y    z, R    f, X, i, n, y    z, X    ta, f, i, n, z    th, f, i, n, z    ph, i
Allowed substitution hints:    ph( y, z, f, n)    ps( y,
z, f, i, n)    ch( y, z, f, i, n)    th( y)    ta( y)    et( y, z, f, i, n)    ze( y, z, f, i, n)    B( y,
f, i, n)    D( y, z, f, n)    K( y, z, f, i, n)

Proof of Theorem bnj1033
StepHypRef Expression
1 bnj1033.1 . . . . 5  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj1033.2 . . . . 5  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj1033.8 . . . . 5  |-  D  =  ( om  \  { (/)
} )
4 bnj1033.9 . . . . 5  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
5 bnj1033.3 . . . . 5  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
61, 2, 3, 4, 5bnj983 33106 . . . 4  |-  ( z  e.  trCl ( X ,  A ,  R )  <->  E. f E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )
7 19.42v 1949 . . . . . . . . . 10  |-  ( E. i ( ( th 
/\  ta )  /\  ( ch  /\  i  e.  n  /\  z  e.  (
f `  i )
) )  <->  ( ( th  /\  ta )  /\  E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) ) )
8 df-3an 975 . . . . . . . . . . 11  |-  ( ( th  /\  ta  /\  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) )  <->  ( ( th  /\  ta )  /\  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) ) )
98exbii 1644 . . . . . . . . . 10  |-  ( E. i ( th  /\  ta  /\  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )  <->  E. i ( ( th 
/\  ta )  /\  ( ch  /\  i  e.  n  /\  z  e.  (
f `  i )
) ) )
10 df-3an 975 . . . . . . . . . 10  |-  ( ( th  /\  ta  /\  E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )  <-> 
( ( th  /\  ta )  /\  E. i
( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) ) )
117, 9, 103bitr4i 277 . . . . . . . . 9  |-  ( E. i ( th  /\  ta  /\  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )  <-> 
( th  /\  ta  /\ 
E. i ( ch 
/\  i  e.  n  /\  z  e.  (
f `  i )
) ) )
1211exbii 1644 . . . . . . . 8  |-  ( E. n E. i ( th  /\  ta  /\  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) )  <->  E. n
( th  /\  ta  /\ 
E. i ( ch 
/\  i  e.  n  /\  z  e.  (
f `  i )
) ) )
13 19.42v 1949 . . . . . . . . 9  |-  ( E. n ( ( th 
/\  ta )  /\  E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )  <-> 
( ( th  /\  ta )  /\  E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) ) )
1410exbii 1644 . . . . . . . . 9  |-  ( E. n ( th  /\  ta  /\  E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) )  <->  E. n
( ( th  /\  ta )  /\  E. i
( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) ) )
15 df-3an 975 . . . . . . . . 9  |-  ( ( th  /\  ta  /\  E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) )  <->  ( ( th  /\  ta )  /\  E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) ) )
1613, 14, 153bitr4i 277 . . . . . . . 8  |-  ( E. n ( th  /\  ta  /\  E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) )  <->  ( th  /\  ta  /\  E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) ) )
1712, 16bitri 249 . . . . . . 7  |-  ( E. n E. i ( th  /\  ta  /\  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) )  <->  ( th  /\  ta  /\  E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) ) )
1817exbii 1644 . . . . . 6  |-  ( E. f E. n E. i ( th  /\  ta  /\  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )  <->  E. f ( th  /\  ta  /\  E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) ) )
19 19.42v 1949 . . . . . . 7  |-  ( E. f ( ( th 
/\  ta )  /\  E. n E. i ( ch 
/\  i  e.  n  /\  z  e.  (
f `  i )
) )  <->  ( ( th  /\  ta )  /\  E. f E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) ) )
2015exbii 1644 . . . . . . 7  |-  ( E. f ( th  /\  ta  /\  E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )  <->  E. f ( ( th 
/\  ta )  /\  E. n E. i ( ch 
/\  i  e.  n  /\  z  e.  (
f `  i )
) ) )
21 df-3an 975 . . . . . . 7  |-  ( ( th  /\  ta  /\  E. f E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )  <-> 
( ( th  /\  ta )  /\  E. f E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) ) )
2219, 20, 213bitr4i 277 . . . . . 6  |-  ( E. f ( th  /\  ta  /\  E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )  <-> 
( th  /\  ta  /\ 
E. f E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) ) )
2318, 22bitri 249 . . . . 5  |-  ( E. f E. n E. i ( th  /\  ta  /\  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )  <-> 
( th  /\  ta  /\ 
E. f E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) ) )
24 bnj255 32855 . . . . . . . 8  |-  ( ( th  /\  ta  /\  ch  /\  ze )  <->  ( th  /\  ta  /\  ( ch 
/\  ze ) ) )
25 bnj1033.7 . . . . . . . . . . 11  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
2625anbi2i 694 . . . . . . . . . 10  |-  ( ( ch  /\  ze )  <->  ( ch  /\  ( i  e.  n  /\  z  e.  ( f `  i
) ) ) )
27 3anass 977 . . . . . . . . . 10  |-  ( ( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) )  <->  ( ch  /\  ( i  e.  n  /\  z  e.  (
f `  i )
) ) )
2826, 27bitr4i 252 . . . . . . . . 9  |-  ( ( ch  /\  ze )  <->  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i
) ) )
29283anbi3i 1189 . . . . . . . 8  |-  ( ( th  /\  ta  /\  ( ch  /\  ze )
)  <->  ( th  /\  ta  /\  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) ) )
3024, 29bitri 249 . . . . . . 7  |-  ( ( th  /\  ta  /\  ch  /\  ze )  <->  ( th  /\  ta  /\  ( ch 
/\  i  e.  n  /\  z  e.  (
f `  i )
) ) )
31303exbii 1646 . . . . . 6  |-  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze ) 
<->  E. f E. n E. i ( th  /\  ta  /\  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) ) )
32 bnj1033.10 . . . . . 6  |-  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)
3331, 32sylbir 213 . . . . 5  |-  ( E. f E. n E. i ( th  /\  ta  /\  ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )  ->  z  e.  B
)
3423, 33sylbir 213 . . . 4  |-  ( ( th  /\  ta  /\  E. f E. n E. i ( ch  /\  i  e.  n  /\  z  e.  ( f `  i ) ) )  ->  z  e.  B
)
356, 34syl3an3b 1266 . . 3  |-  ( ( th  /\  ta  /\  z  e.  trCl ( X ,  A ,  R
) )  ->  z  e.  B )
36353expia 1198 . 2  |-  ( ( th  /\  ta )  ->  ( z  e.  trCl ( X ,  A ,  R )  ->  z  e.  B ) )
3736ssrdv 3510 1  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   {csn 4027   U_ciun 4325   suc csuc 4880    Fn wfn 5583   ` cfv 5588   omcom 6684    /\ w-bnj17 32836    predc-bnj14 32838    FrSe w-bnj15 32842    trClc-bnj18 32844    TrFow-bnj19 32846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-in 3483  df-ss 3490  df-iun 4327  df-fn 5591  df-bnj17 32837  df-bnj18 32845
This theorem is referenced by:  bnj1034  33123
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