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Theorem bnj1128 29807
Description: Technical lemma for bnj69 29827. 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 29769 . . 3  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i ) ) )
7 simp1 1005 . . . . . 6  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  ch )
8 simp2 1006 . . . . . 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 1755 . . . . . . . . . . . . . . 15  |-  F/ j  i  e.  n
11 nfra1 2803 . . . . . . . . . . . . . . . 16  |-  F/ j A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. th )
129, 11nfxfr 1690 . . . . . . . . . . . . . . 15  |-  F/ j ta
13 nfv 1755 . . . . . . . . . . . . . . 15  |-  F/ j ch
1410, 12, 13nf3an 1990 . . . . . . . . . . . . . 14  |-  F/ j ( i  e.  n  /\  ta  /\  ch )
15 nfv 1755 . . . . . . . . . . . . . 14  |-  F/ j ( f `  i
)  C_  A
1614, 15nfim 1980 . . . . . . . . . . . . 13  |-  F/ j ( ( i  e.  n  /\  ta  /\  ch )  ->  ( f `
 i )  C_  A )
1716nfri 1929 . . . . . . . . . . . 12  |-  ( ( ( i  e.  n  /\  ta  /\  ch )  ->  ( f `  i
)  C_  A )  ->  A. j ( ( i  e.  n  /\  ta  /\  ch )  -> 
( f `  i
)  C_  A )
)
183bnj1098 29603 . . . . . . . . . . . . . . . . 17  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
19 simpl 458 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  i  =/=  (/) )
20 simpr1 1011 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  i  e.  n
)
215bnj1232 29623 . . . . . . . . . . . . . . . . . . . 20  |-  ( ch 
->  n  e.  D
)
22213ad2ant3 1028 . . . . . . . . . . . . . . . . . . 19  |-  ( ( i  e.  n  /\  ta  /\  ch )  ->  n  e.  D )
2322adantl 467 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  n  e.  D
)
2419, 20, 233jca 1185 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
) )
2518, 24bnj1101 29604 . . . . . . . . . . . . . . . 16  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
j  e.  n  /\  i  =  suc  j ) )
26 ancl 548 . . . . . . . . . . . . . . . 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 29537 . . . . . . . . . . . . . . 15  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
28 df-3an 984 . . . . . . . . . . . . . . . . 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 313 . . . . . . . . . . . . . . . 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 1712 . . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . . 14  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
32 bnj213 29701 . . . . . . . . . . . . . . . 16  |-  pred (
y ,  A ,  R )  C_  A
3332bnj226 29550 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) 
C_  A
34 simp21 1038 . . . . . . . . . . . . . . . 16  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
i  e.  n )
35 simp3r 1034 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
i  =  suc  j
)
36 biid 239 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  e.  D  <->  n  e.  D )
37 biid 239 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  Fn  n  <->  f  Fn  n )
38 bnj1128.8 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph'  <->  [. j  /  i ]. ph )
39 vex 3083 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  j  e. 
_V
40 sbcg 3365 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( j  e.  _V  ->  ( [. j  /  i ]. ph  <->  ph ) )
4139, 40ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( [. j  /  i ]. ph  <->  ph )
4238, 41bitr2i 253 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  <->  ph' )
43 bnj1128.9 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ps'  <->  [. j  /  i ]. ps )
442, 43bnj1039 29788 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
452, 44bitr4i 255 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ps  <->  ps' )
4636, 37, 42, 45bnj887 29584 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  ( n  e.  D  /\  f  Fn  n  /\  ph'  /\  ps' ) )
47 bnj1128.10 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ch'  <->  [. j  /  i ]. ch )
4838, 43, 5, 47bnj1040 29789 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ch'  <->  (
n  e.  D  /\  f  Fn  n  /\  ph' 
/\  ps' ) )
4946, 5, 483bitr4i 280 . . . . . . . . . . . . . . . . . . . 20  |-  ( ch  <->  ch' )
5048bnj1254 29629 . . . . . . . . . . . . . . . . . . . 20  |-  ( ch'  ->  ps' )
5149, 50sylbi 198 . . . . . . . . . . . . . . . . . . 19  |-  ( ch 
->  ps' )
52513ad2ant3 1028 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  e.  n  /\  ta  /\  ch )  ->  ps' )
53523ad2ant2 1027 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  ->  ps' )
54 simp3l 1033 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
j  e.  n )
55223ad2ant2 1027 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  ->  n  e.  D )
563bnj923 29587 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  D  ->  n  e.  om )
57 elnn 6716 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  n  /\  n  e.  om )  ->  j  e.  om )
5856, 57sylan2 476 . . . . . . . . . . . . . . . . . 18  |-  ( ( j  e.  n  /\  n  e.  D )  ->  j  e.  om )
5954, 55, 58syl2anc 665 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
j  e.  om )
6044bnj589 29728 . . . . . . . . . . . . . . . . . . 19  |-  ( ps'  <->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
61 rsp 2788 . . . . . . . . . . . . . . . . . . 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 ) ) ) )
6260, 61sylbi 198 . . . . . . . . . . . . . . . . . 18  |-  ( ps'  ->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
63 eleq1 2495 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  suc  j  -> 
( i  e.  n  <->  suc  j  e.  n ) )
64 fveq2 5881 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  suc  j  -> 
( f `  i
)  =  ( f `
 suc  j )
)
6564eqeq1d 2424 . . . . . . . . . . . . . . . . . . . 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 ) ) )
6663, 65imbi12d 321 . . . . . . . . . . . . . . . . . . 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 ) ) ) )
6766imbi2d 317 . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
6862, 67syl5ibr 224 . . . . . . . . . . . . . . . . 17  |-  ( i  =  suc  j  -> 
( ps'  ->  ( j  e.  om  ->  ( i  e.  n  ->  ( f `
 i )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) ) )
6935, 53, 59, 68syl3c 63 . . . . . . . . . . . . . . . 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 ) ) )
7034, 69mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
( f `  i
)  =  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) )
7133, 70bnj1262 29630 . . . . . . . . . . . . . 14  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
( f `  i
)  C_  A )
7231, 71bnj1023 29600 . . . . . . . . . . . . 13  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
f `  i )  C_  A )
735bnj1247 29628 . . . . . . . . . . . . . . 15  |-  ( ch 
->  ph )
74733ad2ant3 1028 . . . . . . . . . . . . . 14  |-  ( ( i  e.  n  /\  ta  /\  ch )  ->  ph )
75 bnj213 29701 . . . . . . . . . . . . . . 15  |-  pred ( X ,  A ,  R )  C_  A
76 fveq2 5881 . . . . . . . . . . . . . . . 16  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
771biimpi 197 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
7876, 77sylan9eq 2483 . . . . . . . . . . . . . . 15  |-  ( ( i  =  (/)  /\  ph )  ->  ( f `  i )  =  pred ( X ,  A ,  R ) )
7975, 78bnj1262 29630 . . . . . . . . . . . . . 14  |-  ( ( i  =  (/)  /\  ph )  ->  ( f `  i )  C_  A
)
8074, 79sylan2 476 . . . . . . . . . . . . 13  |-  ( ( i  =  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  ( f `  i )  C_  A
)
8172, 80bnj1109 29606 . . . . . . . . . . . 12  |-  E. j
( ( i  e.  n  /\  ta  /\  ch )  ->  ( f `
 i )  C_  A )
8217, 81bnj1131 29607 . . . . . . . . . . 11  |-  ( ( i  e.  n  /\  ta  /\  ch )  -> 
( f `  i
)  C_  A )
83823expia 1207 . . . . . . . . . 10  |-  ( ( i  e.  n  /\  ta )  ->  ( ch 
->  ( f `  i
)  C_  A )
)
84 bnj1128.6 . . . . . . . . . 10  |-  ( th  <->  ( ch  ->  ( f `  i )  C_  A
) )
8583, 84sylibr 215 . . . . . . . . 9  |-  ( ( i  e.  n  /\  ta )  ->  th )
863, 5, 9, 85bnj1133 29806 . . . . . . . 8  |-  ( ch 
->  A. i  e.  n  th )
8784ralbii 2853 . . . . . . . 8  |-  ( A. i  e.  n  th  <->  A. i  e.  n  ( ch  ->  ( f `  i )  C_  A
) )
8886, 87sylib 199 . . . . . . 7  |-  ( ch 
->  A. i  e.  n  ( ch  ->  ( f `
 i )  C_  A ) )
89 rsp 2788 . . . . . . 7  |-  ( A. i  e.  n  ( ch  ->  ( f `  i )  C_  A
)  ->  ( i  e.  n  ->  ( ch 
->  ( f `  i
)  C_  A )
) )
9088, 89syl 17 . . . . . 6  |-  ( ch 
->  ( i  e.  n  ->  ( ch  ->  (
f `  i )  C_  A ) ) )
917, 8, 7, 90syl3c 63 . . . . 5  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  (
f `  i )  C_  A )
92 simp3 1007 . . . . 5  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  Y  e.  ( f `  i
) )
9391, 92sseldd 3465 . . . 4  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  Y  e.  A )
94932eximi 1702 . . 3  |-  ( E. n E. i ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  E. n E. i  Y  e.  A )
956, 94bnj593 29563 . 2  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i  Y  e.  A )
96 19.9v 1805 . . 3  |-  ( E. f E. n E. i  Y  e.  A  <->  E. n E. i  Y  e.  A )
97 19.9v 1805 . . 3  |-  ( E. n E. i  Y  e.  A  <->  E. i  Y  e.  A )
98 19.9v 1805 . . 3  |-  ( E. i  Y  e.  A  <->  Y  e.  A )
9996, 97, 983bitri 274 . 2  |-  ( E. f E. n E. i  Y  e.  A  <->  Y  e.  A )
10095, 99sylib 199 1  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2407    =/= wne 2614   A.wral 2771   E.wrex 2772   _Vcvv 3080   [.wsbc 3299    \ cdif 3433    C_ wss 3436   (/)c0 3761   {csn 3998   U_ciun 4299   class class class wbr 4423    _E cep 4762   suc csuc 5444    Fn wfn 5596   ` cfv 5601   omcom 6706    /\ w-bnj17 29499    predc-bnj14 29501    trClc-bnj18 29507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-tr 4519  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-iota 5565  df-fn 5604  df-fv 5609  df-om 6707  df-bnj17 29500  df-bnj14 29502  df-bnj18 29508
This theorem is referenced by:  bnj1127  29808
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