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Theorem bnj1128 29020
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
bnj1128.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1128.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1128.3  |-  D  =  ( om  \  { (/)
} )
bnj1128.4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj1128.5  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1128.6  |-  ( th  <->  ( ch  ->  ( f `  i )  C_  A
) )
bnj1128.7  |-  ( ta  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. th ) )
bnj1128.8  |-  ( ph'  <->  [. j  /  i ]. ph )
bnj1128.9  |-  ( ps'  <->  [. j  /  i ]. ps )
bnj1128.10  |-  ( ch'  <->  [. j  /  i ]. ch )
bnj1128.11  |-  ( th'  <->  [. j  / 
i ]. th )
Assertion
Ref Expression
bnj1128  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
Distinct variable groups:    A, f,
i, j, n, y    D, i, j, y    R, f, i, j, n, y   
f, X, i, n, y    f, Y, i, n, y    ch, j    ph, i, y    th, j
Allowed substitution hints:    ph( f, j, n)    ps( y, f, i, j, n)    ch( y,
f, i, n)    th( y,
f, i, n)    ta( y, f, i, j, n)    B( y, f, i, j, n)    D( f, n)    X( j)    Y( j)    ph'( y, f, i, j, n)    ps'( y, f, i, j, n)    ch'( y, f, i, j, n)    th'( y, f, i, j, n)

Proof of Theorem bnj1128
StepHypRef Expression
1 bnj1128.1 . . . 4  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj1128.2 . . . 4  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj1128.3 . . . 4  |-  D  =  ( om  \  { (/)
} )
4 bnj1128.4 . . . 4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
5 bnj1128.5 . . . 4  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
61, 2, 3, 4, 5bnj981 28982 . . 3  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i ) ) )
7 simp1 955 . . . . . 6  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  ch )
8 simp2 956 . . . . . 6  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  i  e.  n )
9 bnj1128.7 . . . . . . . . 9  |-  ( ta  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. th ) )
10 nfv 1605 . . . . . . . . . . . . . . 15  |-  F/ j  i  e.  n
11 nfra1 2593 . . . . . . . . . . . . . . . 16  |-  F/ j A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. th )
129, 11nfxfr 1557 . . . . . . . . . . . . . . 15  |-  F/ j ta
13 nfv 1605 . . . . . . . . . . . . . . 15  |-  F/ j ch
1410, 12, 13nf3an 1774 . . . . . . . . . . . . . 14  |-  F/ j ( i  e.  n  /\  ta  /\  ch )
15 nfv 1605 . . . . . . . . . . . . . 14  |-  F/ j ( f `  i
)  C_  A
1614, 15nfim 1769 . . . . . . . . . . . . 13  |-  F/ j ( ( i  e.  n  /\  ta  /\  ch )  ->  ( f `
 i )  C_  A )
1716nfri 1742 . . . . . . . . . . . 12  |-  ( ( ( i  e.  n  /\  ta  /\  ch )  ->  ( f `  i
)  C_  A )  ->  A. j ( ( i  e.  n  /\  ta  /\  ch )  -> 
( f `  i
)  C_  A )
)
183bnj1098 28815 . . . . . . . . . . . . . . . . 17  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
19 simpl 443 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  i  =/=  (/) )
20 simpr1 961 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  i  e.  n
)
215bnj1232 28836 . . . . . . . . . . . . . . . . . . . 20  |-  ( ch 
->  n  e.  D
)
22213ad2ant3 978 . . . . . . . . . . . . . . . . . . 19  |-  ( ( i  e.  n  /\  ta  /\  ch )  ->  n  e.  D )
2322adantl 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  n  e.  D
)
2419, 20, 233jca 1132 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
) )
2518, 24bnj1101 28816 . . . . . . . . . . . . . . . 16  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
j  e.  n  /\  i  =  suc  j ) )
26 ancl 529 . . . . . . . . . . . . . . . 16  |-  ( ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  ( j  e.  n  /\  i  =  suc  j ) )  ->  ( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\ 
ch ) )  -> 
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  /\  (
j  e.  n  /\  i  =  suc  j ) ) ) )
2725, 26bnj101 28749 . . . . . . . . . . . . . . 15  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
28 df-3an 936 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  <->  ( (
i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
2928imbi2i 303 . . . . . . . . . . . . . . . 16  |-  ( ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch )  /\  (
j  e.  n  /\  i  =  suc  j ) ) )  <->  ( (
i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  ( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\ 
ch ) )  /\  ( j  e.  n  /\  i  =  suc  j ) ) ) )
3029exbii 1569 . . . . . . . . . . . . . . 15  |-  ( E. j ( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\ 
ch ) )  -> 
( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )  <->  E. j ( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\ 
ch ) )  -> 
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  /\  (
j  e.  n  /\  i  =  suc  j ) ) ) )
3127, 30mpbir 200 . . . . . . . . . . . . . 14  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
32 bnj213 28914 . . . . . . . . . . . . . . . 16  |-  pred (
y ,  A ,  R )  C_  A
3332bnj226 28762 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) 
C_  A
34 simp21 988 . . . . . . . . . . . . . . . 16  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
i  e.  n )
35 simp3r 984 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
i  =  suc  j
)
36 biid 227 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  e.  D  <->  n  e.  D )
37 biid 227 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  Fn  n  <->  f  Fn  n )
38 bnj1128.8 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph'  <->  [. j  /  i ]. ph )
39 vex 2791 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  j  e. 
_V
40 nfv 1605 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  F/ i
ph
4140sbcgf 3054 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( j  e.  _V  ->  ( [. j  /  i ]. ph  <->  ph ) )
4239, 41ax-mp 8 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( [. j  /  i ]. ph  <->  ph )
4338, 42bitr2i 241 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  <->  ph' )
44 bnj1128.9 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ps'  <->  [. j  /  i ]. ps )
452, 44bnj1039 29001 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
462, 45bitr4i 243 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ps  <->  ps' )
4736, 37, 43, 46bnj887 28795 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  ( n  e.  D  /\  f  Fn  n  /\  ph'  /\  ps' ) )
48 bnj1128.10 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ch'  <->  [. j  /  i ]. ch )
4938, 44, 5, 48bnj1040 29002 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ch'  <->  (
n  e.  D  /\  f  Fn  n  /\  ph' 
/\  ps' ) )
5047, 5, 493bitr4i 268 . . . . . . . . . . . . . . . . . . . 20  |-  ( ch  <->  ch' )
5149bnj1254 28842 . . . . . . . . . . . . . . . . . . . 20  |-  ( ch'  ->  ps' )
5250, 51sylbi 187 . . . . . . . . . . . . . . . . . . 19  |-  ( ch 
->  ps' )
53523ad2ant3 978 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  e.  n  /\  ta  /\  ch )  ->  ps' )
54533ad2ant2 977 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  ->  ps' )
55 simp3l 983 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
j  e.  n )
56223ad2ant2 977 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  ->  n  e.  D )
573bnj923 28798 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  D  ->  n  e.  om )
58 elnn 4666 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  n  /\  n  e.  om )  ->  j  e.  om )
5957, 58sylan2 460 . . . . . . . . . . . . . . . . . 18  |-  ( ( j  e.  n  /\  n  e.  D )  ->  j  e.  om )
6055, 56, 59syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
j  e.  om )
6145bnj589 28941 . . . . . . . . . . . . . . . . . . 19  |-  ( ps'  <->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
62 rsp 2603 . . . . . . . . . . . . . . . . . . 19  |-  ( A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) )  -> 
( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
6361, 62sylbi 187 . . . . . . . . . . . . . . . . . 18  |-  ( ps'  ->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
64 eleq1 2343 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  suc  j  -> 
( i  e.  n  <->  suc  j  e.  n ) )
65 fveq2 5525 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  suc  j  -> 
( f `  i
)  =  ( f `
 suc  j )
)
6665eqeq1d 2291 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  suc  j  -> 
( ( f `  i )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R )  <->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) )
6764, 66imbi12d 311 . . . . . . . . . . . . . . . . . . 19  |-  ( i  =  suc  j  -> 
( ( i  e.  n  ->  ( f `  i )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) )  <->  ( suc  j  e.  n  ->  ( f `  suc  j
)  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) ) ) )
6867imbi2d 307 . . . . . . . . . . . . . . . . . 18  |-  ( i  =  suc  j  -> 
( ( j  e. 
om  ->  ( i  e.  n  ->  ( f `  i )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )  <-> 
( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) ) )
6963, 68syl5ibr 212 . . . . . . . . . . . . . . . . 17  |-  ( i  =  suc  j  -> 
( ps'  ->  ( j  e.  om  ->  ( i  e.  n  ->  ( f `
 i )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) ) )
7035, 54, 60, 69syl3c 57 . . . . . . . . . . . . . . . 16  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
( i  e.  n  ->  ( f `  i
)  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) ) )
7134, 70mpd 14 . . . . . . . . . . . . . . 15  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
( f `  i
)  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) )
7233, 71bnj1262 28843 . . . . . . . . . . . . . 14  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
( f `  i
)  C_  A )
7331, 72bnj1023 28812 . . . . . . . . . . . . 13  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
f `  i )  C_  A )
745bnj1247 28841 . . . . . . . . . . . . . . 15  |-  ( ch 
->  ph )
75743ad2ant3 978 . . . . . . . . . . . . . 14  |-  ( ( i  e.  n  /\  ta  /\  ch )  ->  ph )
76 bnj213 28914 . . . . . . . . . . . . . . 15  |-  pred ( X ,  A ,  R )  C_  A
77 fveq2 5525 . . . . . . . . . . . . . . . 16  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
781biimpi 186 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
7977, 78sylan9eq 2335 . . . . . . . . . . . . . . 15  |-  ( ( i  =  (/)  /\  ph )  ->  ( f `  i )  =  pred ( X ,  A ,  R ) )
8076, 79bnj1262 28843 . . . . . . . . . . . . . 14  |-  ( ( i  =  (/)  /\  ph )  ->  ( f `  i )  C_  A
)
8175, 80sylan2 460 . . . . . . . . . . . . 13  |-  ( ( i  =  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  ( f `  i )  C_  A
)
8273, 81bnj1109 28818 . . . . . . . . . . . 12  |-  E. j
( ( i  e.  n  /\  ta  /\  ch )  ->  ( f `
 i )  C_  A )
8317, 82bnj1131 28819 . . . . . . . . . . 11  |-  ( ( i  e.  n  /\  ta  /\  ch )  -> 
( f `  i
)  C_  A )
84833expia 1153 . . . . . . . . . 10  |-  ( ( i  e.  n  /\  ta )  ->  ( ch 
->  ( f `  i
)  C_  A )
)
85 bnj1128.6 . . . . . . . . . 10  |-  ( th  <->  ( ch  ->  ( f `  i )  C_  A
) )
8684, 85sylibr 203 . . . . . . . . 9  |-  ( ( i  e.  n  /\  ta )  ->  th )
873, 5, 9, 86bnj1133 29019 . . . . . . . 8  |-  ( ch 
->  A. i  e.  n  th )
8885ralbii 2567 . . . . . . . 8  |-  ( A. i  e.  n  th  <->  A. i  e.  n  ( ch  ->  ( f `  i )  C_  A
) )
8987, 88sylib 188 . . . . . . 7  |-  ( ch 
->  A. i  e.  n  ( ch  ->  ( f `
 i )  C_  A ) )
90 rsp 2603 . . . . . . 7  |-  ( A. i  e.  n  ( ch  ->  ( f `  i )  C_  A
)  ->  ( i  e.  n  ->  ( ch 
->  ( f `  i
)  C_  A )
) )
9189, 90syl 15 . . . . . 6  |-  ( ch 
->  ( i  e.  n  ->  ( ch  ->  (
f `  i )  C_  A ) ) )
927, 8, 7, 91syl3c 57 . . . . 5  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  (
f `  i )  C_  A )
93 simp3 957 . . . . 5  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  Y  e.  ( f `  i
) )
9492, 93sseldd 3181 . . . 4  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  Y  e.  A )
95942eximi 1564 . . 3  |-  ( E. n E. i ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  E. n E. i  Y  e.  A )
966, 95bnj593 28774 . 2  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i  Y  e.  A )
97 19.9v 1663 . . 3  |-  ( E. f E. n E. i  Y  e.  A  <->  E. n E. i  Y  e.  A )
98 19.9v 1663 . . 3  |-  ( E. n E. i  Y  e.  A  <->  E. i  Y  e.  A )
99 19.9v 1663 . . 3  |-  ( E. i  Y  e.  A  <->  Y  e.  A )
10097, 98, 993bitri 262 . 2  |-  ( E. f E. n E. i  Y  e.  A  <->  Y  e.  A )
10196, 100sylib 188 1  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
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    =/= wne 2446   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   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    trClc-bnj18 28719
This theorem is referenced by:  bnj1127  29021
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-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-iota 5219  df-fn 5258  df-fv 5263  df-bnj17 28712  df-bnj14 28714  df-bnj18 28720
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