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Theorem ltsonq 9241
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 9197 . . . . . . 7  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
21adantr 465 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  x  e.  ( N. 
X.  N. ) )
3 xp1st 6708 . . . . . 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 9197 . . . . . . 7  |-  ( y  e.  Q.  ->  y  e.  ( N.  X.  N. ) )
65adantl 466 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  y  e.  ( N. 
X.  N. ) )
7 xp2nd 6709 . . . . . 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 9165 . . . . 5  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  y )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
104, 8, 9syl2anc 661 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
11 xp1st 6708 . . . . . 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 6709 . . . . . 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 9165 . . . . 5  |-  ( ( ( 1st `  y
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  -> 
( ( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
1612, 14, 15syl2anc 661 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
17 ltsopi 9160 . . . . 5  |-  <N  Or  N.
18 sotric 4767 . . . . 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 670 . . . 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 661 . . 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 9215 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  <Q  y  <->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  <N 
( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
22 fveq2 5791 . . . . . . 7  |-  ( x  =  y  ->  ( 1st `  x )  =  ( 1st `  y
) )
23 fveq2 5791 . . . . . . . 8  |-  ( x  =  y  ->  ( 2nd `  x )  =  ( 2nd `  y
) )
2423eqcomd 2459 . . . . . . 7  |-  ( x  =  y  ->  ( 2nd `  y )  =  ( 2nd `  x
) )
2522, 24oveq12d 6210 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) )
26 enqbreq2 9192 . . . . . . . 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 477 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  ~Q  y  <->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
28 enqeq 9206 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  x  ~Q  y )  ->  x  =  y )
29283expia 1190 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  ~Q  y  ->  x  =  y ) )
3027, 29sylbird 235 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( ( 1st `  x )  .N  ( 2nd `  y ) )  =  ( ( 1st `  y )  .N  ( 2nd `  x ) )  ->  x  =  y ) )
3125, 30impbid2 204 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  =  y  <-> 
( ( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
32 ordpinq 9215 . . . . . 6  |-  ( ( y  e.  Q.  /\  x  e.  Q. )  ->  ( y  <Q  x  <->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) )
3332ancoms 453 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( y  <Q  x  <->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) )
3431, 33orbi12d 709 . . . 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 294 . . 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 285 . 2  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  <Q  y  <->  -.  ( x  =  y  \/  y  <Q  x
) ) )
37213adant3 1008 . . . . . 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 9197 . . . . . . . 8  |-  ( z  e.  Q.  ->  z  e.  ( N.  X.  N. ) )
39383ad2ant3 1011 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  z  e.  ( N.  X.  N. ) )
40 xp2nd 6709 . . . . . . 7  |-  ( z  e.  ( N.  X.  N. )  ->  ( 2nd `  z )  e.  N. )
41 ltmpi 9176 . . . . . . 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 20 . . . . . 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 253 . . . . 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 9215 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  <Q  z  <->  ( ( 1st `  y
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  y ) ) ) )
45443adant1 1006 . . . . . 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 1009 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  x  e.  ( N.  X.  N. ) )
47 ltmpi 9176 . . . . . . 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 20 . . . . . 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 253 . . . . 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 710 . . . 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 5801 . . . . . . 7  |-  ( 2nd `  x )  e.  _V
52 fvex 5801 . . . . . . 7  |-  ( 1st `  y )  e.  _V
53 fvex 5801 . . . . . . 7  |-  ( 2nd `  z )  e.  _V
54 mulcompi 9168 . . . . . . 7  |-  ( r  .N  s )  =  ( s  .N  r
)
55 mulasspi 9169 . . . . . . 7  |-  ( ( r  .N  s )  .N  t )  =  ( r  .N  (
s  .N  t ) )
5651, 52, 53, 54, 55caov13 6395 . . . . . 6  |-  ( ( 2nd `  x )  .N  ( ( 1st `  y )  .N  ( 2nd `  z ) ) )  =  ( ( 2nd `  z )  .N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) )
57 fvex 5801 . . . . . . 7  |-  ( 1st `  z )  e.  _V
58 fvex 5801 . . . . . . 7  |-  ( 2nd `  y )  e.  _V
5951, 57, 58, 54, 55caov13 6395 . . . . . 6  |-  ( ( 2nd `  x )  .N  ( ( 1st `  z )  .N  ( 2nd `  y ) ) )  =  ( ( 2nd `  y )  .N  ( ( 1st `  z )  .N  ( 2nd `  x ) ) )
6056, 59breq12i 4401 . . . . 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 5801 . . . . . . 7  |-  ( 1st `  x )  e.  _V
6253, 61, 58, 54, 55caov13 6395 . . . . . 6  |-  ( ( 2nd `  z )  .N  ( ( 1st `  x )  .N  ( 2nd `  y ) ) )  =  ( ( 2nd `  y )  .N  ( ( 1st `  x )  .N  ( 2nd `  z ) ) )
63 ltrelpi 9161 . . . . . . 7  |-  <N  C_  ( N.  X.  N. )
6417, 63sotri 5325 . . . . . 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 4426 . . . . 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 475 . . . 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 228 . . 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 9215 . . . . 5  |-  ( ( x  e.  Q.  /\  z  e.  Q. )  ->  ( x  <Q  z  <->  ( ( 1st `  x
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  x ) ) ) )
69683adant2 1007 . . . 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 1010 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  y  e.  ( N.  X.  N. ) )
71 ltmpi 9176 . . . . 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 20 . . . 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 253 . . 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 234 . 2  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
( x  <Q  y  /\  y  <Q  z )  ->  x  <Q  z
) )
7536, 74isso2i 4773 1  |-  <Q  Or  Q.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4392    Or wor 4740    X. cxp 4938   ` cfv 5518  (class class class)co 6192   1stc1st 6677   2ndc2nd 6678   N.cnpi 9114    .N cmi 9116    <N clti 9117    ~Q ceq 9121   Q.cnq 9122    <Q cltq 9128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-oadd 7026  df-omul 7027  df-er 7203  df-ni 9144  df-mi 9146  df-lti 9147  df-ltpq 9182  df-enq 9183  df-nq 9184  df-ltnq 9190
This theorem is referenced by:  ltbtwnnq  9250  prub  9266  npomex  9268  genpnnp  9277  nqpr  9286  distrlem4pr  9298  prlem934  9305  ltexprlem4  9311  reclem2pr  9320  reclem4pr  9322
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