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Theorem ltsosr 9260
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 9226 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 4294 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  f  <R  [
<. z ,  w >. ]  ~R  ) )
3 eqeq1 2448 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  <->  f  =  [ <. z ,  w >. ]  ~R  ) )
4 breq2 4295 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. x ,  y >. ]  ~R  <->  [ <. z ,  w >. ]  ~R  <R  f
) )
53, 4orbi12d 709 . . . . 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 294 . . . 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 321 . . 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 4295 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( f  <R  [ <. z ,  w >. ]  ~R  <->  f 
<R  g ) )
9 eqeq2 2451 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( f  =  [ <. z ,  w >. ]  ~R  <->  f  =  g ) )
10 breq1 4294 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( [ <. z ,  w >. ]  ~R  <R  f  <-> 
g  <R  f ) )
119, 10orbi12d 709 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( ( f  =  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  f )  <->  ( f  =  g  \/  g  <R  f )
) )
1211notbid 294 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( -.  ( f  =  [ <. z ,  w >. ]  ~R  \/  [
<. z ,  w >. ]  ~R  <R  f )  <->  -.  ( f  =  g  \/  g  <R  f
) ) )
138, 12bibi12d 321 . . 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 9243 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
)
15 addclpr 9186 . . . . . . 7  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  e.  P. )
16 addclpr 9186 . . . . . . 7  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  e.  P. )
17 ltsopr 9200 . . . . . . . 8  |-  <P  Or  P.
18 sotric 4666 . . . . . . . 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 670 . . . . . . 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 477 . . . . . 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 823 . . . . 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 9235 . . . . . . 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 9243 . . . . . . . . 9  |-  ( [
<. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( z  +P.  y ) 
<P  ( w  +P.  x
) )
24 addcompr 9189 . . . . . . . . . 10  |-  ( z  +P.  y )  =  ( y  +P.  z
)
25 addcompr 9189 . . . . . . . . . 10  |-  ( w  +P.  x )  =  ( x  +P.  w
)
2624, 25breq12i 4300 . . . . . . . . 9  |-  ( ( z  +P.  y ) 
<P  ( w  +P.  x
)  <->  ( y  +P.  z )  <P  (
x  +P.  w )
)
2723, 26bitri 249 . . . . . . . 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 709 . . . . . 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 294 . . . . 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 256 . . . 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 257 . . 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 7190 . 2  |-  ( ( f  e.  R.  /\  g  e.  R. )  ->  ( f  <R  g  <->  -.  ( f  =  g  \/  g  <R  f
) ) )
342anbi1d 704 . . . 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 4294 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  f  <R  [
<. v ,  u >. ]  ~R  ) )
3634, 35imbi12d 320 . . 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 4294 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. v ,  u >. ]  ~R  <->  g  <R  [ <. v ,  u >. ]  ~R  ) )
388, 37anbi12d 710 . . . 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 317 . . 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 4295 . . . . 5  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( g  <R  [ <. v ,  u >. ]  ~R  <->  g 
<R  h ) )
4140anbi2d 703 . . . 4  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( ( f  <R 
g  /\  g  <R  [
<. v ,  u >. ]  ~R  )  <->  ( f  <R  g  /\  g  <R  h ) ) )
42 breq2 4295 . . . 4  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( f  <R  [ <. v ,  u >. ]  ~R  <->  f 
<R  h ) )
4341, 42imbi12d 320 . . 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 6115 . . . . . . . . . 10  |-  ( x  +P.  w )  e. 
_V
45 ovex 6115 . . . . . . . . . 10  |-  ( y  +P.  z )  e. 
_V
46 ltapr 9213 . . . . . . . . . 10  |-  ( h  e.  P.  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
47 vex 2974 . . . . . . . . . 10  |-  u  e. 
_V
48 addcompr 9189 . . . . . . . . . 10  |-  ( f  +P.  g )  =  ( g  +P.  f
)
4944, 45, 46, 47, 48caovord2 6274 . . . . . . . . 9  |-  ( u  e.  P.  ->  (
( x  +P.  w
)  <P  ( y  +P.  z )  <->  ( (
x  +P.  w )  +P.  u )  <P  (
( y  +P.  z
)  +P.  u )
) )
50 addasspr 9190 . . . . . . . . . 10  |-  ( ( x  +P.  w )  +P.  u )  =  ( x  +P.  (
w  +P.  u )
)
51 addasspr 9190 . . . . . . . . . 10  |-  ( ( y  +P.  z )  +P.  u )  =  ( y  +P.  (
z  +P.  u )
)
5250, 51breq12i 4300 . . . . . . . . 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 261 . . . . . . . 8  |-  ( u  e.  P.  ->  (
( x  +P.  w
)  <P  ( y  +P.  z )  <->  ( x  +P.  ( w  +P.  u
) )  <P  (
y  +P.  ( z  +P.  u ) ) ) )
5414, 53syl5bb 257 . . . . . . 7  |-  ( u  e.  P.  ->  ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  ( w  +P.  u ) ) 
<P  ( y  +P.  (
z  +P.  u )
) ) )
55 ltsrpr 9243 . . . . . . . 8  |-  ( [
<. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( z  +P.  u )  <P  (
w  +P.  v )
)
56 ltapr 9213 . . . . . . . 8  |-  ( y  e.  P.  ->  (
( z  +P.  u
)  <P  ( w  +P.  v )  <->  ( y  +P.  ( z  +P.  u
) )  <P  (
y  +P.  ( w  +P.  v ) ) ) )
5755, 56syl5bb 257 . . . . . . 7  |-  ( y  e.  P.  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( y  +P.  ( z  +P.  u ) ) 
<P  ( y  +P.  (
w  +P.  v )
) ) )
5854, 57bi2anan9r 869 . . . . . 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 9166 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
6017, 59sotri 5224 . . . . . . 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 9180 . . . . . . . . 9  |-  dom  +P.  =  ( P.  X.  P. )
62 0npr 9160 . . . . . . . . 9  |-  -.  (/)  e.  P.
63 ltapr 9213 . . . . . . . . 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 6253 . . . . . . . 8  |-  ( ( w  +P.  ( x  +P.  u ) ) 
<P  ( w  +P.  (
y  +P.  v )
)  ->  ( x  +P.  u )  <P  (
y  +P.  v )
)
65 vex 2974 . . . . . . . . . 10  |-  x  e. 
_V
66 vex 2974 . . . . . . . . . 10  |-  w  e. 
_V
67 addasspr 9190 . . . . . . . . . 10  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
6865, 66, 47, 48, 67caov12 6290 . . . . . . . . 9  |-  ( x  +P.  ( w  +P.  u ) )  =  ( w  +P.  (
x  +P.  u )
)
69 vex 2974 . . . . . . . . . 10  |-  y  e. 
_V
70 vex 2974 . . . . . . . . . 10  |-  v  e. 
_V
7169, 66, 70, 48, 67caov12 6290 . . . . . . . . 9  |-  ( y  +P.  ( w  +P.  v ) )  =  ( w  +P.  (
y  +P.  v )
)
7268, 71breq12i 4300 . . . . . . . 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 9243 . . . . . . . 8  |-  ( [
<. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( x  +P.  u )  <P  (
y  +P.  v )
)
7464, 72, 733imtr4i 266 . . . . . . 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 228 . . . . 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 745 . . . 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 1007 . . 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 7191 . 2  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
( f  <R  g  /\  g  <R  h )  ->  f  <R  h
) )
8033, 79isso2i 4672 1  |-  <R  Or  R.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3882   class class class wbr 4291    Or wor 4639  (class class class)co 6090   [cec 7098   P.cnp 9025    +P. cpp 9027    <P cltp 9029    ~R cer 9032   R.cnr 9033    <R cltr 9039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-omul 6924  df-er 7100  df-ec 7102  df-qs 7106  df-ni 9040  df-pli 9041  df-mi 9042  df-lti 9043  df-plpq 9076  df-mpq 9077  df-ltpq 9078  df-enq 9079  df-nq 9080  df-erq 9081  df-plq 9082  df-mq 9083  df-1nq 9084  df-rq 9085  df-ltnq 9086  df-np 9149  df-plp 9151  df-ltp 9153  df-enr 9225  df-nr 9226  df-ltr 9229
This theorem is referenced by:  1ne0sr  9262  addgt0sr  9270  sqgt0sr  9272  supsrlem  9277  axpre-lttri  9331  axpre-lttrn  9332
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