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Theorem ltsonq 8593
Description: 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltsonq  |-  <Q  Or  Q.

Proof of Theorem ltsonq
Dummy variables  s 
r  t  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpqn 8549 . . . . . . 7  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
21adantr 451 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  x  e.  ( N. 
X.  N. ) )
3 xp1st 6149 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
42, 3syl 15 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( 1st `  x
)  e.  N. )
5 elpqn 8549 . . . . . . 7  |-  ( y  e.  Q.  ->  y  e.  ( N.  X.  N. ) )
65adantl 452 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  y  e.  ( N. 
X.  N. ) )
7 xp2nd 6150 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( 2nd `  y )  e.  N. )
86, 7syl 15 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( 2nd `  y
)  e.  N. )
9 mulclpi 8517 . . . . 5  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  y )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
104, 8, 9syl2anc 642 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
11 xp1st 6149 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( 1st `  y )  e.  N. )
126, 11syl 15 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( 1st `  y
)  e.  N. )
13 xp2nd 6150 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
142, 13syl 15 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( 2nd `  x
)  e.  N. )
15 mulclpi 8517 . . . . 5  |-  ( ( ( 1st `  y
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  -> 
( ( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
1612, 14, 15syl2anc 642 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
17 ltsopi 8512 . . . . 5  |-  <N  Or  N.
18 sotric 4340 . . . . 5  |-  ( ( 
<N  Or  N.  /\  (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N.  /\  ( ( 1st `  y )  .N  ( 2nd `  x
) )  e.  N. ) )  ->  (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  <N 
( ( 1st `  y
)  .N  ( 2nd `  x ) )  <->  -.  (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  \/  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) ) )
1917, 18mpan 651 . . . 4  |-  ( ( ( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N.  /\  ( ( 1st `  y )  .N  ( 2nd `  x
) )  e.  N. )  ->  ( ( ( 1st `  x )  .N  ( 2nd `  y
) )  <N  (
( 1st `  y
)  .N  ( 2nd `  x ) )  <->  -.  (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  \/  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) ) )
2010, 16, 19syl2anc 642 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( ( 1st `  x )  .N  ( 2nd `  y ) ) 
<N  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <->  -.  (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  \/  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) ) )
21 ordpinq 8567 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  <Q  y  <->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  <N 
( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
22 fveq2 5525 . . . . . . 7  |-  ( x  =  y  ->  ( 1st `  x )  =  ( 1st `  y
) )
23 fveq2 5525 . . . . . . . 8  |-  ( x  =  y  ->  ( 2nd `  x )  =  ( 2nd `  y
) )
2423eqcomd 2288 . . . . . . 7  |-  ( x  =  y  ->  ( 2nd `  y )  =  ( 2nd `  x
) )
2522, 24oveq12d 5876 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) )
26 enqbreq2 8544 . . . . . . . 8  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
x  ~Q  y  <->  ( ( 1st `  x )  .N  ( 2nd `  y
) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
271, 5, 26syl2an 463 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  ~Q  y  <->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
28 enqeq 8558 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  x  ~Q  y )  ->  x  =  y )
29283expia 1153 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  ~Q  y  ->  x  =  y ) )
3027, 29sylbird 226 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( ( 1st `  x )  .N  ( 2nd `  y ) )  =  ( ( 1st `  y )  .N  ( 2nd `  x ) )  ->  x  =  y ) )
3125, 30impbid2 195 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  =  y  <-> 
( ( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
32 ordpinq 8567 . . . . . 6  |-  ( ( y  e.  Q.  /\  x  e.  Q. )  ->  ( y  <Q  x  <->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) )
3332ancoms 439 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( y  <Q  x  <->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) )
3431, 33orbi12d 690 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( x  =  y  \/  y  <Q  x )  <->  ( (
( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  \/  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) ) )
3534notbid 285 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( -.  ( x  =  y  \/  y  <Q  x )  <->  -.  (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  \/  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) ) )
3620, 21, 353bitr4d 276 . 2  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  <Q  y  <->  -.  ( x  =  y  \/  y  <Q  x
) ) )
37213adant3 975 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( ( 1st `  x )  .N  ( 2nd `  y
) )  <N  (
( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
38 elpqn 8549 . . . . . . . 8  |-  ( z  e.  Q.  ->  z  e.  ( N.  X.  N. ) )
39383ad2ant3 978 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  z  e.  ( N.  X.  N. ) )
40 xp2nd 6150 . . . . . . 7  |-  ( z  e.  ( N.  X.  N. )  ->  ( 2nd `  z )  e.  N. )
41 ltmpi 8528 . . . . . . 7  |-  ( ( 2nd `  z )  e.  N.  ->  (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  <N 
( ( 1st `  y
)  .N  ( 2nd `  x ) )  <->  ( ( 2nd `  z )  .N  ( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) 
<N  ( ( 2nd `  z
)  .N  ( ( 1st `  y )  .N  ( 2nd `  x
) ) ) ) )
4239, 40, 413syl 18 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  <N 
( ( 1st `  y
)  .N  ( 2nd `  x ) )  <->  ( ( 2nd `  z )  .N  ( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) 
<N  ( ( 2nd `  z
)  .N  ( ( 1st `  y )  .N  ( 2nd `  x
) ) ) ) )
4337, 42bitrd 244 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( ( 2nd `  z )  .N  ( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) 
<N  ( ( 2nd `  z
)  .N  ( ( 1st `  y )  .N  ( 2nd `  x
) ) ) ) )
44 ordpinq 8567 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  <Q  z  <->  ( ( 1st `  y
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  y ) ) ) )
45443adant1 973 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
y  <Q  z  <->  ( ( 1st `  y )  .N  ( 2nd `  z
) )  <N  (
( 1st `  z
)  .N  ( 2nd `  y ) ) ) )
4613ad2ant1 976 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  x  e.  ( N.  X.  N. ) )
47 ltmpi 8528 . . . . . . 7  |-  ( ( 2nd `  x )  e.  N.  ->  (
( ( 1st `  y
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  y ) )  <->  ( ( 2nd `  x )  .N  ( ( 1st `  y
)  .N  ( 2nd `  z ) ) ) 
<N  ( ( 2nd `  x
)  .N  ( ( 1st `  z )  .N  ( 2nd `  y
) ) ) ) )
4846, 13, 473syl 18 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
( ( 1st `  y
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  y ) )  <->  ( ( 2nd `  x )  .N  ( ( 1st `  y
)  .N  ( 2nd `  z ) ) ) 
<N  ( ( 2nd `  x
)  .N  ( ( 1st `  z )  .N  ( 2nd `  y
) ) ) ) )
4945, 48bitrd 244 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
y  <Q  z  <->  ( ( 2nd `  x )  .N  ( ( 1st `  y
)  .N  ( 2nd `  z ) ) ) 
<N  ( ( 2nd `  x
)  .N  ( ( 1st `  z )  .N  ( 2nd `  y
) ) ) ) )
5043, 49anbi12d 691 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
( x  <Q  y  /\  y  <Q  z )  <-> 
( ( ( 2nd `  z )  .N  (
( 1st `  x
)  .N  ( 2nd `  y ) ) ) 
<N  ( ( 2nd `  z
)  .N  ( ( 1st `  y )  .N  ( 2nd `  x
) ) )  /\  ( ( 2nd `  x
)  .N  ( ( 1st `  y )  .N  ( 2nd `  z
) ) )  <N 
( ( 2nd `  x
)  .N  ( ( 1st `  z )  .N  ( 2nd `  y
) ) ) ) ) )
51 fvex 5539 . . . . . . 7  |-  ( 2nd `  x )  e.  _V
52 fvex 5539 . . . . . . 7  |-  ( 1st `  y )  e.  _V
53 fvex 5539 . . . . . . 7  |-  ( 2nd `  z )  e.  _V
54 mulcompi 8520 . . . . . . 7  |-  ( r  .N  s )  =  ( s  .N  r
)
55 mulasspi 8521 . . . . . . 7  |-  ( ( r  .N  s )  .N  t )  =  ( r  .N  (
s  .N  t ) )
5651, 52, 53, 54, 55caov13 6050 . . . . . 6  |-  ( ( 2nd `  x )  .N  ( ( 1st `  y )  .N  ( 2nd `  z ) ) )  =  ( ( 2nd `  z )  .N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) )
57 fvex 5539 . . . . . . 7  |-  ( 1st `  z )  e.  _V
58 fvex 5539 . . . . . . 7  |-  ( 2nd `  y )  e.  _V
5951, 57, 58, 54, 55caov13 6050 . . . . . 6  |-  ( ( 2nd `  x )  .N  ( ( 1st `  z )  .N  ( 2nd `  y ) ) )  =  ( ( 2nd `  y )  .N  ( ( 1st `  z )  .N  ( 2nd `  x ) ) )
6056, 59breq12i 4032 . . . . 5  |-  ( ( ( 2nd `  x
)  .N  ( ( 1st `  y )  .N  ( 2nd `  z
) ) )  <N 
( ( 2nd `  x
)  .N  ( ( 1st `  z )  .N  ( 2nd `  y
) ) )  <->  ( ( 2nd `  z )  .N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) 
<N  ( ( 2nd `  y
)  .N  ( ( 1st `  z )  .N  ( 2nd `  x
) ) ) )
61 fvex 5539 . . . . . . 7  |-  ( 1st `  x )  e.  _V
6253, 61, 58, 54, 55caov13 6050 . . . . . 6  |-  ( ( 2nd `  z )  .N  ( ( 1st `  x )  .N  ( 2nd `  y ) ) )  =  ( ( 2nd `  y )  .N  ( ( 1st `  x )  .N  ( 2nd `  z ) ) )
63 ltrelpi 8513 . . . . . . 7  |-  <N  C_  ( N.  X.  N. )
6417, 63sotri 5070 . . . . . 6  |-  ( ( ( ( 2nd `  z
)  .N  ( ( 1st `  x )  .N  ( 2nd `  y
) ) )  <N 
( ( 2nd `  z
)  .N  ( ( 1st `  y )  .N  ( 2nd `  x
) ) )  /\  ( ( 2nd `  z
)  .N  ( ( 1st `  y )  .N  ( 2nd `  x
) ) )  <N 
( ( 2nd `  y
)  .N  ( ( 1st `  z )  .N  ( 2nd `  x
) ) ) )  ->  ( ( 2nd `  z )  .N  (
( 1st `  x
)  .N  ( 2nd `  y ) ) ) 
<N  ( ( 2nd `  y
)  .N  ( ( 1st `  z )  .N  ( 2nd `  x
) ) ) )
6562, 64syl5eqbrr 4057 . . . . 5  |-  ( ( ( ( 2nd `  z
)  .N  ( ( 1st `  x )  .N  ( 2nd `  y
) ) )  <N 
( ( 2nd `  z
)  .N  ( ( 1st `  y )  .N  ( 2nd `  x
) ) )  /\  ( ( 2nd `  z
)  .N  ( ( 1st `  y )  .N  ( 2nd `  x
) ) )  <N 
( ( 2nd `  y
)  .N  ( ( 1st `  z )  .N  ( 2nd `  x
) ) ) )  ->  ( ( 2nd `  y )  .N  (
( 1st `  x
)  .N  ( 2nd `  z ) ) ) 
<N  ( ( 2nd `  y
)  .N  ( ( 1st `  z )  .N  ( 2nd `  x
) ) ) )
6660, 65sylan2b 461 . . . 4  |-  ( ( ( ( 2nd `  z
)  .N  ( ( 1st `  x )  .N  ( 2nd `  y
) ) )  <N 
( ( 2nd `  z
)  .N  ( ( 1st `  y )  .N  ( 2nd `  x
) ) )  /\  ( ( 2nd `  x
)  .N  ( ( 1st `  y )  .N  ( 2nd `  z
) ) )  <N 
( ( 2nd `  x
)  .N  ( ( 1st `  z )  .N  ( 2nd `  y
) ) ) )  ->  ( ( 2nd `  y )  .N  (
( 1st `  x
)  .N  ( 2nd `  z ) ) ) 
<N  ( ( 2nd `  y
)  .N  ( ( 1st `  z )  .N  ( 2nd `  x
) ) ) )
6750, 66syl6bi 219 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
( x  <Q  y  /\  y  <Q  z )  ->  ( ( 2nd `  y )  .N  (
( 1st `  x
)  .N  ( 2nd `  z ) ) ) 
<N  ( ( 2nd `  y
)  .N  ( ( 1st `  z )  .N  ( 2nd `  x
) ) ) ) )
68 ordpinq 8567 . . . . 5  |-  ( ( x  e.  Q.  /\  z  e.  Q. )  ->  ( x  <Q  z  <->  ( ( 1st `  x
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  x ) ) ) )
69683adant2 974 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  z  <->  ( ( 1st `  x )  .N  ( 2nd `  z
) )  <N  (
( 1st `  z
)  .N  ( 2nd `  x ) ) ) )
7053ad2ant2 977 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  y  e.  ( N.  X.  N. ) )
71 ltmpi 8528 . . . . 5  |-  ( ( 2nd `  y )  e.  N.  ->  (
( ( 1st `  x
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  x ) )  <->  ( ( 2nd `  y )  .N  ( ( 1st `  x
)  .N  ( 2nd `  z ) ) ) 
<N  ( ( 2nd `  y
)  .N  ( ( 1st `  z )  .N  ( 2nd `  x
) ) ) ) )
7270, 7, 713syl 18 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
( ( 1st `  x
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  x ) )  <->  ( ( 2nd `  y )  .N  ( ( 1st `  x
)  .N  ( 2nd `  z ) ) ) 
<N  ( ( 2nd `  y
)  .N  ( ( 1st `  z )  .N  ( 2nd `  x
) ) ) ) )
7369, 72bitrd 244 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  z  <->  ( ( 2nd `  y )  .N  ( ( 1st `  x
)  .N  ( 2nd `  z ) ) ) 
<N  ( ( 2nd `  y
)  .N  ( ( 1st `  z )  .N  ( 2nd `  x
) ) ) ) )
7467, 73sylibrd 225 . 2  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
( x  <Q  y  /\  y  <Q  z )  ->  x  <Q  z
) )
7536, 74isso2i 4346 1  |-  <Q  Or  Q.
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023    Or wor 4313    X. cxp 4687   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   N.cnpi 8466    .N cmi 8468    <N clti 8469    ~Q ceq 8473   Q.cnq 8474    <Q cltq 8480
This theorem is referenced by:  ltbtwnnq  8602  prub  8618  npomex  8620  genpnnp  8629  nqpr  8638  distrlem4pr  8650  prlem934  8657  ltexprlem4  8663  reclem2pr  8672  reclem4pr  8674
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-pow 4188  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-reu 2550  df-rmo 2551  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-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  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-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-mi 8498  df-lti 8499  df-ltpq 8534  df-enq 8535  df-nq 8536  df-ltnq 8542
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