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Theorem ltsonq 8847
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 8803 . . . . . . 7  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
21adantr 453 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  x  e.  ( N. 
X.  N. ) )
3 xp1st 6377 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
42, 3syl 16 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( 1st `  x
)  e.  N. )
5 elpqn 8803 . . . . . . 7  |-  ( y  e.  Q.  ->  y  e.  ( N.  X.  N. ) )
65adantl 454 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  y  e.  ( N. 
X.  N. ) )
7 xp2nd 6378 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( 2nd `  y )  e.  N. )
86, 7syl 16 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( 2nd `  y
)  e.  N. )
9 mulclpi 8771 . . . . 5  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  y )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
104, 8, 9syl2anc 644 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
11 xp1st 6377 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( 1st `  y )  e.  N. )
126, 11syl 16 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( 1st `  y
)  e.  N. )
13 xp2nd 6378 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
142, 13syl 16 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( 2nd `  x
)  e.  N. )
15 mulclpi 8771 . . . . 5  |-  ( ( ( 1st `  y
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  -> 
( ( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
1612, 14, 15syl2anc 644 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
17 ltsopi 8766 . . . . 5  |-  <N  Or  N.
18 sotric 4530 . . . . 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 653 . . . 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 644 . . 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 8821 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  <Q  y  <->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  <N 
( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
22 fveq2 5729 . . . . . . 7  |-  ( x  =  y  ->  ( 1st `  x )  =  ( 1st `  y
) )
23 fveq2 5729 . . . . . . . 8  |-  ( x  =  y  ->  ( 2nd `  x )  =  ( 2nd `  y
) )
2423eqcomd 2442 . . . . . . 7  |-  ( x  =  y  ->  ( 2nd `  y )  =  ( 2nd `  x
) )
2522, 24oveq12d 6100 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) )
26 enqbreq2 8798 . . . . . . . 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 465 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  ~Q  y  <->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
28 enqeq 8812 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  x  ~Q  y )  ->  x  =  y )
29283expia 1156 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  ~Q  y  ->  x  =  y ) )
3027, 29sylbird 228 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( ( 1st `  x )  .N  ( 2nd `  y ) )  =  ( ( 1st `  y )  .N  ( 2nd `  x ) )  ->  x  =  y ) )
3125, 30impbid2 197 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  =  y  <-> 
( ( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
32 ordpinq 8821 . . . . . 6  |-  ( ( y  e.  Q.  /\  x  e.  Q. )  ->  ( y  <Q  x  <->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) )
3332ancoms 441 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( y  <Q  x  <->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) )
3431, 33orbi12d 692 . . . 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 287 . . 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 278 . 2  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  <Q  y  <->  -.  ( x  =  y  \/  y  <Q  x
) ) )
37213adant3 978 . . . . . 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 8803 . . . . . . . 8  |-  ( z  e.  Q.  ->  z  e.  ( N.  X.  N. ) )
39383ad2ant3 981 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  z  e.  ( N.  X.  N. ) )
40 xp2nd 6378 . . . . . . 7  |-  ( z  e.  ( N.  X.  N. )  ->  ( 2nd `  z )  e.  N. )
41 ltmpi 8782 . . . . . . 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 19 . . . . . 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 246 . . . . 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 8821 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  <Q  z  <->  ( ( 1st `  y
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  y ) ) ) )
45443adant1 976 . . . . . 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 979 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  x  e.  ( N.  X.  N. ) )
47 ltmpi 8782 . . . . . . 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 19 . . . . . 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 246 . . . . 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 693 . . . 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 5743 . . . . . . 7  |-  ( 2nd `  x )  e.  _V
52 fvex 5743 . . . . . . 7  |-  ( 1st `  y )  e.  _V
53 fvex 5743 . . . . . . 7  |-  ( 2nd `  z )  e.  _V
54 mulcompi 8774 . . . . . . 7  |-  ( r  .N  s )  =  ( s  .N  r
)
55 mulasspi 8775 . . . . . . 7  |-  ( ( r  .N  s )  .N  t )  =  ( r  .N  (
s  .N  t ) )
5651, 52, 53, 54, 55caov13 6278 . . . . . 6  |-  ( ( 2nd `  x )  .N  ( ( 1st `  y )  .N  ( 2nd `  z ) ) )  =  ( ( 2nd `  z )  .N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) )
57 fvex 5743 . . . . . . 7  |-  ( 1st `  z )  e.  _V
58 fvex 5743 . . . . . . 7  |-  ( 2nd `  y )  e.  _V
5951, 57, 58, 54, 55caov13 6278 . . . . . 6  |-  ( ( 2nd `  x )  .N  ( ( 1st `  z )  .N  ( 2nd `  y ) ) )  =  ( ( 2nd `  y )  .N  ( ( 1st `  z )  .N  ( 2nd `  x ) ) )
6056, 59breq12i 4222 . . . . 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 5743 . . . . . . 7  |-  ( 1st `  x )  e.  _V
6253, 61, 58, 54, 55caov13 6278 . . . . . 6  |-  ( ( 2nd `  z )  .N  ( ( 1st `  x )  .N  ( 2nd `  y ) ) )  =  ( ( 2nd `  y )  .N  ( ( 1st `  x )  .N  ( 2nd `  z ) ) )
63 ltrelpi 8767 . . . . . . 7  |-  <N  C_  ( N.  X.  N. )
6417, 63sotri 5262 . . . . . 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 4247 . . . . 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 463 . . . 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 221 . . 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 8821 . . . . 5  |-  ( ( x  e.  Q.  /\  z  e.  Q. )  ->  ( x  <Q  z  <->  ( ( 1st `  x
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  x ) ) ) )
69683adant2 977 . . . 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 980 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  y  e.  ( N.  X.  N. ) )
71 ltmpi 8782 . . . . 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 19 . . . 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 246 . . 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 227 . 2  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
( x  <Q  y  /\  y  <Q  z )  ->  x  <Q  z
) )
7536, 74isso2i 4536 1  |-  <Q  Or  Q.
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4213    Or wor 4503    X. cxp 4877   ` cfv 5455  (class class class)co 6082   1stc1st 6348   2ndc2nd 6349   N.cnpi 8720    .N cmi 8722    <N clti 8723    ~Q ceq 8727   Q.cnq 8728    <Q cltq 8734
This theorem is referenced by:  ltbtwnnq  8856  prub  8872  npomex  8874  genpnnp  8883  nqpr  8892  distrlem4pr  8904  prlem934  8911  ltexprlem4  8917  reclem2pr  8926  reclem4pr  8928
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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-recs 6634  df-rdg 6669  df-oadd 6729  df-omul 6730  df-er 6906  df-ni 8750  df-mi 8752  df-lti 8753  df-ltpq 8788  df-enq 8789  df-nq 8790  df-ltnq 8796
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