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Theorem bnj1128 29361
Description: Technical lemma for bnj69 29381. 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 29323 . . 3  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i ) ) )
7 simp1 958 . . . . . 6  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  ch )
8 simp2 959 . . . . . 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 1630 . . . . . . . . . . . . . . 15  |-  F/ j  i  e.  n
11 nfra1 2758 . . . . . . . . . . . . . . . 16  |-  F/ j A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. th )
129, 11nfxfr 1580 . . . . . . . . . . . . . . 15  |-  F/ j ta
13 nfv 1630 . . . . . . . . . . . . . . 15  |-  F/ j ch
1410, 12, 13nf3an 1850 . . . . . . . . . . . . . 14  |-  F/ j ( i  e.  n  /\  ta  /\  ch )
15 nfv 1630 . . . . . . . . . . . . . 14  |-  F/ j ( f `  i
)  C_  A
1614, 15nfim 1833 . . . . . . . . . . . . 13  |-  F/ j ( ( i  e.  n  /\  ta  /\  ch )  ->  ( f `
 i )  C_  A )
1716nfri 1779 . . . . . . . . . . . 12  |-  ( ( ( i  e.  n  /\  ta  /\  ch )  ->  ( f `  i
)  C_  A )  ->  A. j ( ( i  e.  n  /\  ta  /\  ch )  -> 
( f `  i
)  C_  A )
)
183bnj1098 29156 . . . . . . . . . . . . . . . . 17  |-  E. j
( ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
)  ->  ( j  e.  n  /\  i  =  suc  j ) )
19 simpl 445 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  i  =/=  (/) )
20 simpr1 964 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  i  e.  n
)
215bnj1232 29177 . . . . . . . . . . . . . . . . . . . 20  |-  ( ch 
->  n  e.  D
)
22213ad2ant3 981 . . . . . . . . . . . . . . . . . . 19  |-  ( ( i  e.  n  /\  ta  /\  ch )  ->  n  e.  D )
2322adantl 454 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  n  e.  D
)
2419, 20, 233jca 1135 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  ( i  =/=  (/)  /\  i  e.  n  /\  n  e.  D
) )
2518, 24bnj1101 29157 . . . . . . . . . . . . . . . 16  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
j  e.  n  /\  i  =  suc  j ) )
26 ancl 531 . . . . . . . . . . . . . . . 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 29090 . . . . . . . . . . . . . . 15  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
28 df-3an 939 . . . . . . . . . . . . . . . . 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 305 . . . . . . . . . . . . . . . 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 1593 . . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . . 14  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )
32 bnj213 29255 . . . . . . . . . . . . . . . 16  |-  pred (
y ,  A ,  R )  C_  A
3332bnj226 29103 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( f `  j
)  pred ( y ,  A ,  R ) 
C_  A
34 simp21 991 . . . . . . . . . . . . . . . 16  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
i  e.  n )
35 simp3r 987 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
i  =  suc  j
)
36 biid 229 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  e.  D  <->  n  e.  D )
37 biid 229 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  Fn  n  <->  f  Fn  n )
38 bnj1128.8 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph'  <->  [. j  /  i ]. ph )
39 vex 2961 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  j  e. 
_V
40 sbcg 3228 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( j  e.  _V  ->  ( [. j  /  i ]. ph  <->  ph ) )
4139, 40ax-mp 8 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( [. j  /  i ]. ph  <->  ph )
4238, 41bitr2i 243 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  <->  ph' )
43 bnj1128.9 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ps'  <->  [. j  /  i ]. ps )
442, 43bnj1039 29342 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
452, 44bitr4i 245 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ps  <->  ps' )
4636, 37, 42, 45bnj887 29136 . . . . . . . . . . . . . . . . . . . . 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 29343 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ch'  <->  (
n  e.  D  /\  f  Fn  n  /\  ph' 
/\  ps' ) )
4946, 5, 483bitr4i 270 . . . . . . . . . . . . . . . . . . . 20  |-  ( ch  <->  ch' )
5048bnj1254 29183 . . . . . . . . . . . . . . . . . . . 20  |-  ( ch'  ->  ps' )
5149, 50sylbi 189 . . . . . . . . . . . . . . . . . . 19  |-  ( ch 
->  ps' )
52513ad2ant3 981 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  e.  n  /\  ta  /\  ch )  ->  ps' )
53523ad2ant2 980 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  ->  ps' )
54 simp3l 986 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
j  e.  n )
55223ad2ant2 980 . . . . . . . . . . . . . . . . . 18  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  ->  n  e.  D )
563bnj923 29139 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  D  ->  n  e.  om )
57 elnn 4857 . . . . . . . . . . . . . . . . . . 19  |-  ( ( j  e.  n  /\  n  e.  om )  ->  j  e.  om )
5856, 57sylan2 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( j  e.  n  /\  n  e.  D )  ->  j  e.  om )
5954, 55, 58syl2anc 644 . . . . . . . . . . . . . . . . 17  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
j  e.  om )
6044bnj589 29282 . . . . . . . . . . . . . . . . . . 19  |-  ( ps'  <->  A. j  e.  om  ( suc  j  e.  n  ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j )  pred (
y ,  A ,  R ) ) )
61 rsp 2768 . . . . . . . . . . . . . . . . . . 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 189 . . . . . . . . . . . . . . . . . 18  |-  ( ps'  ->  ( j  e.  om  ->  ( suc  j  e.  n  ->  ( f `  suc  j )  = 
U_ y  e.  ( f `  j ) 
pred ( y ,  A ,  R ) ) ) )
63 eleq1 2498 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  suc  j  -> 
( i  e.  n  <->  suc  j  e.  n ) )
64 fveq2 5730 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  suc  j  -> 
( f `  i
)  =  ( f `
 suc  j )
)
6564eqeq1d 2446 . . . . . . . . . . . . . . . . . . . 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 313 . . . . . . . . . . . . . . . . . . 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 309 . . . . . . . . . . . . . . . . . 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 214 . . . . . . . . . . . . . . . . 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 60 . . . . . . . . . . . . . . . 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 29184 . . . . . . . . . . . . . 14  |-  ( ( i  =/=  (/)  /\  (
i  e.  n  /\  ta  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) )  -> 
( f `  i
)  C_  A )
7231, 71bnj1023 29153 . . . . . . . . . . . . 13  |-  E. j
( ( i  =/=  (/)  /\  ( i  e.  n  /\  ta  /\  ch ) )  ->  (
f `  i )  C_  A )
735bnj1247 29182 . . . . . . . . . . . . . . 15  |-  ( ch 
->  ph )
74733ad2ant3 981 . . . . . . . . . . . . . 14  |-  ( ( i  e.  n  /\  ta  /\  ch )  ->  ph )
75 bnj213 29255 . . . . . . . . . . . . . . 15  |-  pred ( X ,  A ,  R )  C_  A
76 fveq2 5730 . . . . . . . . . . . . . . . 16  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
771biimpi 188 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
7876, 77sylan9eq 2490 . . . . . . . . . . . . . . 15  |-  ( ( i  =  (/)  /\  ph )  ->  ( f `  i )  =  pred ( X ,  A ,  R ) )
7975, 78bnj1262 29184 . . . . . . . . . . . . . 14  |-  ( ( i  =  (/)  /\  ph )  ->  ( f `  i )  C_  A
)
8074, 79sylan2 462 . . . . . . . . . . . . 13  |-  ( ( i  =  (/)  /\  (
i  e.  n  /\  ta  /\  ch ) )  ->  ( f `  i )  C_  A
)
8172, 80bnj1109 29159 . . . . . . . . . . . 12  |-  E. j
( ( i  e.  n  /\  ta  /\  ch )  ->  ( f `
 i )  C_  A )
8217, 81bnj1131 29160 . . . . . . . . . . 11  |-  ( ( i  e.  n  /\  ta  /\  ch )  -> 
( f `  i
)  C_  A )
83823expia 1156 . . . . . . . . . 10  |-  ( ( i  e.  n  /\  ta )  ->  ( ch 
->  ( f `  i
)  C_  A )
)
84 bnj1128.6 . . . . . . . . . 10  |-  ( th  <->  ( ch  ->  ( f `  i )  C_  A
) )
8583, 84sylibr 205 . . . . . . . . 9  |-  ( ( i  e.  n  /\  ta )  ->  th )
863, 5, 9, 85bnj1133 29360 . . . . . . . 8  |-  ( ch 
->  A. i  e.  n  th )
8784ralbii 2731 . . . . . . . 8  |-  ( A. i  e.  n  th  <->  A. i  e.  n  ( ch  ->  ( f `  i )  C_  A
) )
8886, 87sylib 190 . . . . . . 7  |-  ( ch 
->  A. i  e.  n  ( ch  ->  ( f `
 i )  C_  A ) )
89 rsp 2768 . . . . . . 7  |-  ( A. i  e.  n  ( ch  ->  ( f `  i )  C_  A
)  ->  ( i  e.  n  ->  ( ch 
->  ( f `  i
)  C_  A )
) )
9088, 89syl 16 . . . . . 6  |-  ( ch 
->  ( i  e.  n  ->  ( ch  ->  (
f `  i )  C_  A ) ) )
917, 8, 7, 90syl3c 60 . . . . 5  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  (
f `  i )  C_  A )
92 simp3 960 . . . . 5  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  Y  e.  ( f `  i
) )
9391, 92sseldd 3351 . . . 4  |-  ( ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  Y  e.  A )
94932eximi 1587 . . 3  |-  ( E. n E. i ( ch  /\  i  e.  n  /\  Y  e.  ( f `  i
) )  ->  E. n E. i  Y  e.  A )
956, 94bnj593 29115 . 2  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i  Y  e.  A )
96 19.9v 1677 . . 3  |-  ( E. f E. n E. i  Y  e.  A  <->  E. n E. i  Y  e.  A )
97 19.9v 1677 . . 3  |-  ( E. n E. i  Y  e.  A  <->  E. i  Y  e.  A )
98 19.9v 1677 . . 3  |-  ( E. i  Y  e.  A  <->  Y  e.  A )
9996, 97, 983bitri 264 . 2  |-  ( E. f E. n E. i  Y  e.  A  <->  Y  e.  A )
10095, 99sylib 190 1  |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958   [.wsbc 3163    \ cdif 3319    C_ wss 3322   (/)c0 3630   {csn 3816   U_ciun 4095   class class class wbr 4214    _E cep 4494   suc csuc 4585   omcom 4847    Fn wfn 5451   ` cfv 5456    /\ w-bnj17 29052    predc-bnj14 29054    trClc-bnj18 29060
This theorem is referenced by:  bnj1127  29362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-iota 5420  df-fn 5459  df-fv 5464  df-bnj17 29053  df-bnj14 29055  df-bnj18 29061
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