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Theorem ltsosr 8961
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 8927 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 4207 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  f  <R  [
<. z ,  w >. ]  ~R  ) )
3 eqeq1 2441 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  <->  f  =  [ <. z ,  w >. ]  ~R  ) )
4 breq2 4208 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. x ,  y >. ]  ~R  <->  [ <. z ,  w >. ]  ~R  <R  f
) )
53, 4orbi12d 691 . . . . 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 286 . . . 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 313 . . 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 4208 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( f  <R  [ <. z ,  w >. ]  ~R  <->  f 
<R  g ) )
9 eqeq2 2444 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( f  =  [ <. z ,  w >. ]  ~R  <->  f  =  g ) )
10 breq1 4207 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( [ <. z ,  w >. ]  ~R  <R  f  <-> 
g  <R  f ) )
119, 10orbi12d 691 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( ( f  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  f )  <->  ( f  =  g  \/  g  <R  f )
) )
1211notbid 286 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( -.  ( f  =  [ <. z ,  w >. ]  ~R  \/  [
<. z ,  w >. ]  ~R  <R  f )  <->  -.  ( f  =  g  \/  g  <R  f
) ) )
138, 12bibi12d 313 . . 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 8944 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
)
15 addclpr 8887 . . . . . . 7  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  e.  P. )
16 addclpr 8887 . . . . . . 7  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  e.  P. )
17 ltsopr 8901 . . . . . . . 8  |-  <P  Or  P.
18 sotric 4521 . . . . . . . 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 652 . . . . . . 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 464 . . . . . 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 801 . . . . 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 8936 . . . . . . 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 8944 . . . . . . . . 9  |-  ( [
<. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( z  +P.  y ) 
<P  ( w  +P.  x
) )
24 addcompr 8890 . . . . . . . . . 10  |-  ( z  +P.  y )  =  ( y  +P.  z
)
25 addcompr 8890 . . . . . . . . . 10  |-  ( w  +P.  x )  =  ( x  +P.  w
)
2624, 25breq12i 4213 . . . . . . . . 9  |-  ( ( z  +P.  y ) 
<P  ( w  +P.  x
)  <->  ( y  +P.  z )  <P  (
x  +P.  w )
)
2723, 26bitri 241 . . . . . . . 8  |-  ( [
<. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( y  +P.  z ) 
<P  ( x  +P.  w
) )
2827a1i 11 . . . . . . 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 691 . . . . . 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 286 . . . . 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 248 . . . 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 249 . . 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 6987 . 2  |-  ( ( f  e.  R.  /\  g  e.  R. )  ->  ( f  <R  g  <->  -.  ( f  =  g  \/  g  <R  f
) ) )
342anbi1d 686 . . . 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 4207 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  f  <R  [
<. v ,  u >. ]  ~R  ) )
3634, 35imbi12d 312 . . 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 4207 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. v ,  u >. ]  ~R  <->  g  <R  [ <. v ,  u >. ]  ~R  ) )
388, 37anbi12d 692 . . . 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 309 . . 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 4208 . . . . 5  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( g  <R  [ <. v ,  u >. ]  ~R  <->  g 
<R  h ) )
4140anbi2d 685 . . . 4  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( ( f  <R 
g  /\  g  <R  [
<. v ,  u >. ]  ~R  )  <->  ( f  <R  g  /\  g  <R  h ) ) )
42 breq2 4208 . . . 4  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( f  <R  [ <. v ,  u >. ]  ~R  <->  f 
<R  h ) )
4341, 42imbi12d 312 . . 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 6098 . . . . . . . . . 10  |-  ( x  +P.  w )  e. 
_V
45 ovex 6098 . . . . . . . . . 10  |-  ( y  +P.  z )  e. 
_V
46 ltapr 8914 . . . . . . . . . 10  |-  ( h  e.  P.  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
47 vex 2951 . . . . . . . . . 10  |-  u  e. 
_V
48 addcompr 8890 . . . . . . . . . 10  |-  ( f  +P.  g )  =  ( g  +P.  f
)
4944, 45, 46, 47, 48caovord2 6251 . . . . . . . . 9  |-  ( u  e.  P.  ->  (
( x  +P.  w
)  <P  ( y  +P.  z )  <->  ( (
x  +P.  w )  +P.  u )  <P  (
( y  +P.  z
)  +P.  u )
) )
50 addasspr 8891 . . . . . . . . . 10  |-  ( ( x  +P.  w )  +P.  u )  =  ( x  +P.  (
w  +P.  u )
)
51 addasspr 8891 . . . . . . . . . 10  |-  ( ( y  +P.  z )  +P.  u )  =  ( y  +P.  (
z  +P.  u )
)
5250, 51breq12i 4213 . . . . . . . . 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 253 . . . . . . . 8  |-  ( u  e.  P.  ->  (
( x  +P.  w
)  <P  ( y  +P.  z )  <->  ( x  +P.  ( w  +P.  u
) )  <P  (
y  +P.  ( z  +P.  u ) ) ) )
5414, 53syl5bb 249 . . . . . . 7  |-  ( u  e.  P.  ->  ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  ( w  +P.  u ) ) 
<P  ( y  +P.  (
z  +P.  u )
) ) )
55 ltsrpr 8944 . . . . . . . 8  |-  ( [
<. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( z  +P.  u )  <P  (
w  +P.  v )
)
56 ltapr 8914 . . . . . . . 8  |-  ( y  e.  P.  ->  (
( z  +P.  u
)  <P  ( w  +P.  v )  <->  ( y  +P.  ( z  +P.  u
) )  <P  (
y  +P.  ( w  +P.  v ) ) ) )
5755, 56syl5bb 249 . . . . . . 7  |-  ( y  e.  P.  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( y  +P.  ( z  +P.  u ) ) 
<P  ( y  +P.  (
w  +P.  v )
) ) )
5854, 57bi2anan9r 845 . . . . . 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 8867 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
6017, 59sotri 5253 . . . . . . 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 8881 . . . . . . . . 9  |-  dom  +P.  =  ( P.  X.  P. )
62 0npr 8861 . . . . . . . . 9  |-  -.  (/)  e.  P.
63 ltapr 8914 . . . . . . . . 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 6230 . . . . . . . 8  |-  ( ( w  +P.  ( x  +P.  u ) ) 
<P  ( w  +P.  (
y  +P.  v )
)  ->  ( x  +P.  u )  <P  (
y  +P.  v )
)
65 vex 2951 . . . . . . . . . 10  |-  x  e. 
_V
66 vex 2951 . . . . . . . . . 10  |-  w  e. 
_V
67 addasspr 8891 . . . . . . . . . 10  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
6865, 66, 47, 48, 67caov12 6267 . . . . . . . . 9  |-  ( x  +P.  ( w  +P.  u ) )  =  ( w  +P.  (
x  +P.  u )
)
69 vex 2951 . . . . . . . . . 10  |-  y  e. 
_V
70 vex 2951 . . . . . . . . . 10  |-  v  e. 
_V
7169, 66, 70, 48, 67caov12 6267 . . . . . . . . 9  |-  ( y  +P.  ( w  +P.  v ) )  =  ( w  +P.  (
y  +P.  v )
)
7268, 71breq12i 4213 . . . . . . . 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 8944 . . . . . . . 8  |-  ( [
<. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( x  +P.  u )  <P  (
y  +P.  v )
)
7464, 72, 733imtr4i 258 . . . . . . 7  |-  ( ( x  +P.  ( w  +P.  u ) ) 
<P  ( y  +P.  (
w  +P.  v )
)  ->  [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )
7560, 74syl 16 . . . . . 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 220 . . . . 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 727 . . . 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 976 . . 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 6988 . 2  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
( f  <R  g  /\  g  <R  h )  ->  f  <R  h
) )
8033, 79isso2i 4527 1  |-  <R  Or  R.
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204    Or wor 4494  (class class class)co 6073   [cec 6895   P.cnp 8726    +P. cpp 8728    <P cltp 8730    ~R cer 8733   R.cnr 8734    <R cltr 8740
This theorem is referenced by:  1ne0sr  8963  addgt0sr  8971  sqgt0sr  8973  supsrlem  8978  axpre-lttri  9032  axpre-lttrn  9033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-ec 6899  df-qs 6903  df-ni 8741  df-pli 8742  df-mi 8743  df-lti 8744  df-plpq 8777  df-mpq 8778  df-ltpq 8779  df-enq 8780  df-nq 8781  df-erq 8782  df-plq 8783  df-mq 8784  df-1nq 8785  df-rq 8786  df-ltnq 8787  df-np 8850  df-plp 8852  df-ltp 8854  df-enr 8926  df-nr 8927  df-ltr 8930
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