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Theorem ltsosr 8716
Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltsosr  |-  <R  Or  R.

Proof of Theorem ltsosr
Dummy variables  x  y  z  w  v  u  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 8682 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 4026 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  f  <R  [
<. z ,  w >. ]  ~R  ) )
3 eqeq1 2289 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  <->  f  =  [ <. z ,  w >. ]  ~R  ) )
4 breq2 4027 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. x ,  y >. ]  ~R  <->  [ <. z ,  w >. ]  ~R  <R  f
) )
53, 4orbi12d 690 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( ( [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  ) 
<->  ( f  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  f ) ) )
65notbid 285 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( -.  ( [
<. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  ) 
<->  -.  ( f  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  f )
) )
72, 6bibi12d 312 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  -.  ( [ <. x ,  y
>. ]  ~R  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y
>. ]  ~R  ) )  <-> 
( f  <R  [ <. z ,  w >. ]  ~R  <->  -.  ( f  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  f ) ) ) )
8 breq2 4027 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( f  <R  [ <. z ,  w >. ]  ~R  <->  f 
<R  g ) )
9 eqeq2 2292 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( f  =  [ <. z ,  w >. ]  ~R  <->  f  =  g ) )
10 breq1 4026 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( [ <. z ,  w >. ]  ~R  <R  f  <-> 
g  <R  f ) )
119, 10orbi12d 690 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( ( f  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  f )  <->  ( f  =  g  \/  g  <R  f )
) )
1211notbid 285 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( -.  ( f  =  [ <. z ,  w >. ]  ~R  \/  [
<. z ,  w >. ]  ~R  <R  f )  <->  -.  ( f  =  g  \/  g  <R  f
) ) )
138, 12bibi12d 312 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( ( f  <R  [ <. z ,  w >. ]  ~R  <->  -.  (
f  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  f
) )  <->  ( f  <R  g  <->  -.  ( f  =  g  \/  g  <R  f ) ) ) )
14 ltsrpr 8699 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
)
15 addclpr 8642 . . . . . . 7  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  e.  P. )
16 addclpr 8642 . . . . . . 7  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  e.  P. )
17 ltsopr 8656 . . . . . . . 8  |-  <P  Or  P.
18 sotric 4340 . . . . . . . 8  |-  ( ( 
<P  Or  P.  /\  (
( x  +P.  w
)  e.  P.  /\  ( y  +P.  z
)  e.  P. )
)  ->  ( (
x  +P.  w )  <P  ( y  +P.  z
)  <->  -.  ( (
x  +P.  w )  =  ( y  +P.  z )  \/  (
y  +P.  z )  <P  ( x  +P.  w
) ) ) )
1917, 18mpan 651 . . . . . . 7  |-  ( ( ( x  +P.  w
)  e.  P.  /\  ( y  +P.  z
)  e.  P. )  ->  ( ( x  +P.  w )  <P  (
y  +P.  z )  <->  -.  ( ( x  +P.  w )  =  ( y  +P.  z )  \/  ( y  +P.  z )  <P  (
x  +P.  w )
) ) )
2015, 16, 19syl2an 463 . . . . . 6  |-  ( ( ( x  e.  P.  /\  w  e.  P. )  /\  ( y  e.  P.  /\  z  e.  P. )
)  ->  ( (
x  +P.  w )  <P  ( y  +P.  z
)  <->  -.  ( (
x  +P.  w )  =  ( y  +P.  z )  \/  (
y  +P.  z )  <P  ( x  +P.  w
) ) ) )
2120an42s 800 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  +P.  w )  <P  ( y  +P.  z
)  <->  -.  ( (
x  +P.  w )  =  ( y  +P.  z )  \/  (
y  +P.  z )  <P  ( x  +P.  w
) ) ) )
22 enreceq 8691 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  =  ( y  +P.  z ) ) )
23 ltsrpr 8699 . . . . . . . . 9  |-  ( [
<. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( z  +P.  y ) 
<P  ( w  +P.  x
) )
24 addcompr 8645 . . . . . . . . . 10  |-  ( z  +P.  y )  =  ( y  +P.  z
)
25 addcompr 8645 . . . . . . . . . 10  |-  ( w  +P.  x )  =  ( x  +P.  w
)
2624, 25breq12i 4032 . . . . . . . . 9  |-  ( ( z  +P.  y ) 
<P  ( w  +P.  x
)  <->  ( y  +P.  z )  <P  (
x  +P.  w )
)
2723, 26bitri 240 . . . . . . . 8  |-  ( [
<. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( y  +P.  z ) 
<P  ( x  +P.  w
) )
2827a1i 10 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( y  +P.  z ) 
<P  ( x  +P.  w
) ) )
2922, 28orbi12d 690 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( ( [ <. x ,  y
>. ]  ~R  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y
>. ]  ~R  )  <->  ( (
x  +P.  w )  =  ( y  +P.  z )  \/  (
y  +P.  z )  <P  ( x  +P.  w
) ) ) )
3029notbid 285 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( -.  ( [ <. x ,  y
>. ]  ~R  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y
>. ]  ~R  )  <->  -.  (
( x  +P.  w
)  =  ( y  +P.  z )  \/  ( y  +P.  z
)  <P  ( x  +P.  w ) ) ) )
3121, 30bitr4d 247 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  +P.  w )  <P  ( y  +P.  z
)  <->  -.  ( [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  ) ) )
3214, 31syl5bb 248 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  -.  ( [ <. x ,  y
>. ]  ~R  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y
>. ]  ~R  ) ) )
331, 7, 13, 322ecoptocl 6749 . 2  |-  ( ( f  e.  R.  /\  g  e.  R. )  ->  ( f  <R  g  <->  -.  ( f  =  g  \/  g  <R  f
) ) )
342anbi1d 685 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  /\  [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  <->  ( f  <R  [ <. z ,  w >. ]  ~R  /\  [
<. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )
) )
35 breq1 4026 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  f  <R  [
<. v ,  u >. ]  ~R  ) )
3634, 35imbi12d 311 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( ( ( [
<. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  /\  [
<. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  ->  [ <. x ,  y
>. ]  ~R  <R  [ <. v ,  u >. ]  ~R  ) 
<->  ( ( f  <R  [ <. z ,  w >. ]  ~R  /\  [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  ->  f  <R  [ <. v ,  u >. ]  ~R  )
) )
37 breq1 4026 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. v ,  u >. ]  ~R  <->  g  <R  [ <. v ,  u >. ]  ~R  ) )
388, 37anbi12d 691 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( ( f  <R  [ <. z ,  w >. ]  ~R  /\  [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  <->  ( f  <R  g  /\  g  <R  [ <. v ,  u >. ]  ~R  )
) )
3938imbi1d 308 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( ( ( f 
<R  [ <. z ,  w >. ]  ~R  /\  [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  ->  f  <R  [ <. v ,  u >. ]  ~R  )  <->  ( ( f  <R  g  /\  g  <R  [ <. v ,  u >. ]  ~R  )  ->  f  <R  [ <. v ,  u >. ]  ~R  ) ) )
40 breq2 4027 . . . . 5  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( g  <R  [ <. v ,  u >. ]  ~R  <->  g 
<R  h ) )
4140anbi2d 684 . . . 4  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( ( f  <R 
g  /\  g  <R  [
<. v ,  u >. ]  ~R  )  <->  ( f  <R  g  /\  g  <R  h ) ) )
42 breq2 4027 . . . 4  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( f  <R  [ <. v ,  u >. ]  ~R  <->  f 
<R  h ) )
4341, 42imbi12d 311 . . 3  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( ( ( f 
<R  g  /\  g  <R  [ <. v ,  u >. ]  ~R  )  -> 
f  <R  [ <. v ,  u >. ]  ~R  )  <->  ( ( f  <R  g  /\  g  <R  h )  ->  f  <R  h
) ) )
44 ovex 5883 . . . . . . . . . 10  |-  ( x  +P.  w )  e. 
_V
45 ovex 5883 . . . . . . . . . 10  |-  ( y  +P.  z )  e. 
_V
46 ltapr 8669 . . . . . . . . . 10  |-  ( h  e.  P.  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
47 vex 2791 . . . . . . . . . 10  |-  u  e. 
_V
48 addcompr 8645 . . . . . . . . . 10  |-  ( f  +P.  g )  =  ( g  +P.  f
)
4944, 45, 46, 47, 48caovord2 6032 . . . . . . . . 9  |-  ( u  e.  P.  ->  (
( x  +P.  w
)  <P  ( y  +P.  z )  <->  ( (
x  +P.  w )  +P.  u )  <P  (
( y  +P.  z
)  +P.  u )
) )
50 addasspr 8646 . . . . . . . . . 10  |-  ( ( x  +P.  w )  +P.  u )  =  ( x  +P.  (
w  +P.  u )
)
51 addasspr 8646 . . . . . . . . . 10  |-  ( ( y  +P.  z )  +P.  u )  =  ( y  +P.  (
z  +P.  u )
)
5250, 51breq12i 4032 . . . . . . . . 9  |-  ( ( ( x  +P.  w
)  +P.  u )  <P  ( ( y  +P.  z )  +P.  u
)  <->  ( x  +P.  ( w  +P.  u ) )  <P  ( y  +P.  ( z  +P.  u
) ) )
5349, 52syl6bb 252 . . . . . . . 8  |-  ( u  e.  P.  ->  (
( x  +P.  w
)  <P  ( y  +P.  z )  <->  ( x  +P.  ( w  +P.  u
) )  <P  (
y  +P.  ( z  +P.  u ) ) ) )
5414, 53syl5bb 248 . . . . . . 7  |-  ( u  e.  P.  ->  ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  ( w  +P.  u ) ) 
<P  ( y  +P.  (
z  +P.  u )
) ) )
55 ltsrpr 8699 . . . . . . . 8  |-  ( [
<. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( z  +P.  u )  <P  (
w  +P.  v )
)
56 ltapr 8669 . . . . . . . 8  |-  ( y  e.  P.  ->  (
( z  +P.  u
)  <P  ( w  +P.  v )  <->  ( y  +P.  ( z  +P.  u
) )  <P  (
y  +P.  ( w  +P.  v ) ) ) )
5755, 56syl5bb 248 . . . . . . 7  |-  ( y  e.  P.  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( y  +P.  ( z  +P.  u ) ) 
<P  ( y  +P.  (
w  +P.  v )
) ) )
5854, 57bi2anan9r 844 . . . . . 6  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  /\  [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  <->  ( ( x  +P.  (
w  +P.  u )
)  <P  ( y  +P.  ( z  +P.  u
) )  /\  (
y  +P.  ( z  +P.  u ) )  <P 
( y  +P.  (
w  +P.  v )
) ) ) )
59 ltrelpr 8622 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
6017, 59sotri 5070 . . . . . . 7  |-  ( ( ( x  +P.  (
w  +P.  u )
)  <P  ( y  +P.  ( z  +P.  u
) )  /\  (
y  +P.  ( z  +P.  u ) )  <P 
( y  +P.  (
w  +P.  v )
) )  ->  (
x  +P.  ( w  +P.  u ) )  <P 
( y  +P.  (
w  +P.  v )
) )
61 dmplp 8636 . . . . . . . . 9  |-  dom  +P.  =  ( P.  X.  P. )
62 0npr 8616 . . . . . . . . 9  |-  -.  (/)  e.  P.
63 ltapr 8669 . . . . . . . . 9  |-  ( w  e.  P.  ->  (
( x  +P.  u
)  <P  ( y  +P.  v )  <->  ( w  +P.  ( x  +P.  u
) )  <P  (
w  +P.  ( y  +P.  v ) ) ) )
6461, 59, 62, 63ndmovordi 6011 . . . . . . . 8  |-  ( ( w  +P.  ( x  +P.  u ) ) 
<P  ( w  +P.  (
y  +P.  v )
)  ->  ( x  +P.  u )  <P  (
y  +P.  v )
)
65 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
66 vex 2791 . . . . . . . . . 10  |-  w  e. 
_V
67 addasspr 8646 . . . . . . . . . 10  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
6865, 66, 47, 48, 67caov12 6048 . . . . . . . . 9  |-  ( x  +P.  ( w  +P.  u ) )  =  ( w  +P.  (
x  +P.  u )
)
69 vex 2791 . . . . . . . . . 10  |-  y  e. 
_V
70 vex 2791 . . . . . . . . . 10  |-  v  e. 
_V
7169, 66, 70, 48, 67caov12 6048 . . . . . . . . 9  |-  ( y  +P.  ( w  +P.  v ) )  =  ( w  +P.  (
y  +P.  v )
)
7268, 71breq12i 4032 . . . . . . . 8  |-  ( ( x  +P.  ( w  +P.  u ) ) 
<P  ( y  +P.  (
w  +P.  v )
)  <->  ( w  +P.  ( x  +P.  u ) )  <P  ( w  +P.  ( y  +P.  v
) ) )
73 ltsrpr 8699 . . . . . . . 8  |-  ( [
<. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( x  +P.  u )  <P  (
y  +P.  v )
)
7464, 72, 733imtr4i 257 . . . . . . 7  |-  ( ( x  +P.  ( w  +P.  u ) ) 
<P  ( y  +P.  (
w  +P.  v )
)  ->  [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )
7560, 74syl 15 . . . . . 6  |-  ( ( ( x  +P.  (
w  +P.  u )
)  <P  ( y  +P.  ( z  +P.  u
) )  /\  (
y  +P.  ( z  +P.  u ) )  <P 
( y  +P.  (
w  +P.  v )
) )  ->  [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )
7658, 75syl6bi 219 . . . . 5  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  /\  [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  ->  [ <. x ,  y
>. ]  ~R  <R  [ <. v ,  u >. ]  ~R  ) )
7776ad2ant2l 726 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  /\ 
[ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  ->  [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  ) )
78773adant2 974 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  /\ 
[ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  ->  [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  ) )
791, 36, 39, 43, 783ecoptocl 6750 . 2  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
( f  <R  g  /\  g  <R  h )  ->  f  <R  h
) )
8033, 79isso2i 4346 1  |-  <R  Or  R.
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023    Or wor 4313  (class class class)co 5858   [cec 6658   P.cnp 8481    +P. cpp 8483    <P cltp 8485    ~R cer 8488   R.cnr 8489    <R cltr 8495
This theorem is referenced by:  1ne0sr  8718  addgt0sr  8726  sqgt0sr  8728  supsrlem  8733  axpre-lttri  8787  axpre-lttrn  8788
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  ax-inf2 7342
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-int 3863  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-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-rq 8541  df-ltnq 8542  df-np 8605  df-plp 8607  df-ltp 8609  df-enr 8681  df-nr 8682  df-ltr 8685
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