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Theorem ltsonq 8884
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 8840 . . . . . . 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 6412 . . . . . 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 8840 . . . . . . 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 6413 . . . . . 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 8808 . . . . 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 6412 . . . . . 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 6413 . . . . . 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 8808 . . . . 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 8803 . . . . 5  |-  <N  Or  N.
18 sotric 4564 . . . . 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 8858 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  <Q  y  <->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  <N 
( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
22 fveq2 5763 . . . . . . 7  |-  ( x  =  y  ->  ( 1st `  x )  =  ( 1st `  y
) )
23 fveq2 5763 . . . . . . . 8  |-  ( x  =  y  ->  ( 2nd `  x )  =  ( 2nd `  y
) )
2423eqcomd 2448 . . . . . . 7  |-  ( x  =  y  ->  ( 2nd `  y )  =  ( 2nd `  x
) )
2522, 24oveq12d 6135 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) )
26 enqbreq2 8835 . . . . . . . 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 8849 . . . . . . . 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 8858 . . . . . 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 8840 . . . . . . . 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 6413 . . . . . . 7  |-  ( z  e.  ( N.  X.  N. )  ->  ( 2nd `  z )  e.  N. )
41 ltmpi 8819 . . . . . . 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 8858 . . . . . . 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 8819 . . . . . . 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 5773 . . . . . . 7  |-  ( 2nd `  x )  e.  _V
52 fvex 5773 . . . . . . 7  |-  ( 1st `  y )  e.  _V
53 fvex 5773 . . . . . . 7  |-  ( 2nd `  z )  e.  _V
54 mulcompi 8811 . . . . . . 7  |-  ( r  .N  s )  =  ( s  .N  r
)
55 mulasspi 8812 . . . . . . 7  |-  ( ( r  .N  s )  .N  t )  =  ( r  .N  (
s  .N  t ) )
5651, 52, 53, 54, 55caov13 6313 . . . . . 6  |-  ( ( 2nd `  x )  .N  ( ( 1st `  y )  .N  ( 2nd `  z ) ) )  =  ( ( 2nd `  z )  .N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) )
57 fvex 5773 . . . . . . 7  |-  ( 1st `  z )  e.  _V
58 fvex 5773 . . . . . . 7  |-  ( 2nd `  y )  e.  _V
5951, 57, 58, 54, 55caov13 6313 . . . . . 6  |-  ( ( 2nd `  x )  .N  ( ( 1st `  z )  .N  ( 2nd `  y ) ) )  =  ( ( 2nd `  y )  .N  ( ( 1st `  z )  .N  ( 2nd `  x ) ) )
6056, 59breq12i 4252 . . . . 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 5773 . . . . . . 7  |-  ( 1st `  x )  e.  _V
6253, 61, 58, 54, 55caov13 6313 . . . . . 6  |-  ( ( 2nd `  z )  .N  ( ( 1st `  x )  .N  ( 2nd `  y ) ) )  =  ( ( 2nd `  y )  .N  ( ( 1st `  x )  .N  ( 2nd `  z ) ) )
63 ltrelpi 8804 . . . . . . 7  |-  <N  C_  ( N.  X.  N. )
6417, 63sotri 5296 . . . . . 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 4277 . . . . 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 8858 . . . . 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 8819 . . . . 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 4570 1  |-  <Q  Or  Q.
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   class class class wbr 4243    Or wor 4537    X. cxp 4911   ` cfv 5489  (class class class)co 6117   1stc1st 6383   2ndc2nd 6384   N.cnpi 8757    .N cmi 8759    <N clti 8760    ~Q ceq 8764   Q.cnq 8765    <Q cltq 8771
This theorem is referenced by:  ltbtwnnq  8893  prub  8909  npomex  8911  genpnnp  8920  nqpr  8929  distrlem4pr  8941  prlem934  8948  ltexprlem4  8954  reclem2pr  8963  reclem4pr  8965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
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 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-recs 6669  df-rdg 6704  df-oadd 6764  df-omul 6765  df-er 6941  df-ni 8787  df-mi 8789  df-lti 8790  df-ltpq 8825  df-enq 8826  df-nq 8827  df-ltnq 8833
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