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Theorem ltsonq 9258
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 9214 . . . . . . 7  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
21adantr 463 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  x  e.  ( N. 
X.  N. ) )
3 xp1st 6729 . . . . . 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 9214 . . . . . . 7  |-  ( y  e.  Q.  ->  y  e.  ( N.  X.  N. ) )
65adantl 464 . . . . . 6  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  y  e.  ( N. 
X.  N. ) )
7 xp2nd 6730 . . . . . 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 9182 . . . . 5  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  y )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
104, 8, 9syl2anc 659 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  e. 
N. )
11 xp1st 6729 . . . . . 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 6730 . . . . . 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 9182 . . . . 5  |-  ( ( ( 1st `  y
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  -> 
( ( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
1612, 14, 15syl2anc 659 . . . 4  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  e. 
N. )
17 ltsopi 9177 . . . . 5  |-  <N  Or  N.
18 sotric 4740 . . . . 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 668 . . . 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 659 . . 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 9232 . . 3  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  <Q  y  <->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  <N 
( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
22 fveq2 5774 . . . . . . 7  |-  ( x  =  y  ->  ( 1st `  x )  =  ( 1st `  y
) )
23 fveq2 5774 . . . . . . . 8  |-  ( x  =  y  ->  ( 2nd `  x )  =  ( 2nd `  y
) )
2423eqcomd 2390 . . . . . . 7  |-  ( x  =  y  ->  ( 2nd `  y )  =  ( 2nd `  x
) )
2522, 24oveq12d 6214 . . . . . 6  |-  ( x  =  y  ->  (
( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) )
26 enqbreq2 9209 . . . . . . . 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 475 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  ~Q  y  <->  ( ( 1st `  x
)  .N  ( 2nd `  y ) )  =  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
28 enqeq 9223 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  x  ~Q  y )  ->  x  =  y )
29283expia 1196 . . . . . . 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 9232 . . . . . 6  |-  ( ( y  e.  Q.  /\  x  e.  Q. )  ->  ( y  <Q  x  <->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) )
3332ancoms 451 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( y  <Q  x  <->  ( ( 1st `  y
)  .N  ( 2nd `  x ) )  <N 
( ( 1st `  x
)  .N  ( 2nd `  y ) ) ) )
3431, 33orbi12d 707 . . . 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 292 . . 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 1014 . . . . . 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 9214 . . . . . . . 8  |-  ( z  e.  Q.  ->  z  e.  ( N.  X.  N. ) )
39383ad2ant3 1017 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  z  e.  ( N.  X.  N. ) )
40 xp2nd 6730 . . . . . . 7  |-  ( z  e.  ( N.  X.  N. )  ->  ( 2nd `  z )  e.  N. )
41 ltmpi 9193 . . . . . . 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 9232 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  <Q  z  <->  ( ( 1st `  y
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  y ) ) ) )
45443adant1 1012 . . . . . 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 1015 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  x  e.  ( N.  X.  N. ) )
47 ltmpi 9193 . . . . . . 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 708 . . . 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 5784 . . . . . . 7  |-  ( 2nd `  x )  e.  _V
52 fvex 5784 . . . . . . 7  |-  ( 1st `  y )  e.  _V
53 fvex 5784 . . . . . . 7  |-  ( 2nd `  z )  e.  _V
54 mulcompi 9185 . . . . . . 7  |-  ( r  .N  s )  =  ( s  .N  r
)
55 mulasspi 9186 . . . . . . 7  |-  ( ( r  .N  s )  .N  t )  =  ( r  .N  (
s  .N  t ) )
5651, 52, 53, 54, 55caov13 6404 . . . . . 6  |-  ( ( 2nd `  x )  .N  ( ( 1st `  y )  .N  ( 2nd `  z ) ) )  =  ( ( 2nd `  z )  .N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) )
57 fvex 5784 . . . . . . 7  |-  ( 1st `  z )  e.  _V
58 fvex 5784 . . . . . . 7  |-  ( 2nd `  y )  e.  _V
5951, 57, 58, 54, 55caov13 6404 . . . . . 6  |-  ( ( 2nd `  x )  .N  ( ( 1st `  z )  .N  ( 2nd `  y ) ) )  =  ( ( 2nd `  y )  .N  ( ( 1st `  z )  .N  ( 2nd `  x ) ) )
6056, 59breq12i 4376 . . . . 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 5784 . . . . . . 7  |-  ( 1st `  x )  e.  _V
6253, 61, 58, 54, 55caov13 6404 . . . . . 6  |-  ( ( 2nd `  z )  .N  ( ( 1st `  x )  .N  ( 2nd `  y ) ) )  =  ( ( 2nd `  y )  .N  ( ( 1st `  x )  .N  ( 2nd `  z ) ) )
63 ltrelpi 9178 . . . . . . 7  |-  <N  C_  ( N.  X.  N. )
6417, 63sotri 5307 . . . . . 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 4401 . . . . 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 473 . . . 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 9232 . . . . 5  |-  ( ( x  e.  Q.  /\  z  e.  Q. )  ->  ( x  <Q  z  <->  ( ( 1st `  x
)  .N  ( 2nd `  z ) )  <N 
( ( 1st `  z
)  .N  ( 2nd `  x ) ) ) )
69683adant2 1013 . . . 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 1016 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  y  e.  ( N.  X.  N. ) )
71 ltmpi 9193 . . . . 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 4746 1  |-  <Q  Or  Q.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   class class class wbr 4367    Or wor 4713    X. cxp 4911   ` cfv 5496  (class class class)co 6196   1stc1st 6697   2ndc2nd 6698   N.cnpi 9133    .N cmi 9135    <N clti 9136    ~Q ceq 9140   Q.cnq 9141    <Q cltq 9147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  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 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-oadd 7052  df-omul 7053  df-er 7229  df-ni 9161  df-mi 9163  df-lti 9164  df-ltpq 9199  df-enq 9200  df-nq 9201  df-ltnq 9207
This theorem is referenced by:  ltbtwnnq  9267  prub  9283  npomex  9285  genpnnp  9294  nqpr  9303  distrlem4pr  9315  prlem934  9322  ltexprlem4  9328  reclem2pr  9337  reclem4pr  9339
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