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Theorem ltsosr 9462
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 9425 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 4445 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  f  <R  [
<. z ,  w >. ]  ~R  ) )
3 eqeq1 2466 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  <->  f  =  [ <. z ,  w >. ]  ~R  ) )
4 breq2 4446 . . . . . 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 4446 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( f  <R  [ <. z ,  w >. ]  ~R  <->  f 
<R  g ) )
9 eqeq2 2477 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( f  =  [ <. z ,  w >. ]  ~R  <->  f  =  g ) )
10 breq1 4445 . . . . . 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 9445 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
)
15 addclpr 9387 . . . . . . 7  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  e.  P. )
16 addclpr 9387 . . . . . . 7  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  e.  P. )
17 ltsopr 9401 . . . . . . . 8  |-  <P  Or  P.
18 sotric 4821 . . . . . . . 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 824 . . . . 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 9434 . . . . . . 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 9445 . . . . . . . . 9  |-  ( [
<. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( z  +P.  y ) 
<P  ( w  +P.  x
) )
24 addcompr 9390 . . . . . . . . . 10  |-  ( z  +P.  y )  =  ( y  +P.  z
)
25 addcompr 9390 . . . . . . . . . 10  |-  ( w  +P.  x )  =  ( x  +P.  w
)
2624, 25breq12i 4451 . . . . . . . . 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 7394 . 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 4445 . . . 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 4445 . . . . 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 4446 . . . . 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 4446 . . . 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 6302 . . . . . . . . . 10  |-  ( x  +P.  w )  e. 
_V
45 ovex 6302 . . . . . . . . . 10  |-  ( y  +P.  z )  e. 
_V
46 ltapr 9414 . . . . . . . . . 10  |-  ( h  e.  P.  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
47 vex 3111 . . . . . . . . . 10  |-  u  e. 
_V
48 addcompr 9390 . . . . . . . . . 10  |-  ( f  +P.  g )  =  ( g  +P.  f
)
4944, 45, 46, 47, 48caovord2 6464 . . . . . . . . 9  |-  ( u  e.  P.  ->  (
( x  +P.  w
)  <P  ( y  +P.  z )  <->  ( (
x  +P.  w )  +P.  u )  <P  (
( y  +P.  z
)  +P.  u )
) )
50 addasspr 9391 . . . . . . . . . 10  |-  ( ( x  +P.  w )  +P.  u )  =  ( x  +P.  (
w  +P.  u )
)
51 addasspr 9391 . . . . . . . . . 10  |-  ( ( y  +P.  z )  +P.  u )  =  ( y  +P.  (
z  +P.  u )
)
5250, 51breq12i 4451 . . . . . . . . 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 9445 . . . . . . . 8  |-  ( [
<. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( z  +P.  u )  <P  (
w  +P.  v )
)
56 ltapr 9414 . . . . . . . 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 870 . . . . . 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 9367 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
6017, 59sotri 5387 . . . . . . 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 9381 . . . . . . . . 9  |-  dom  +P.  =  ( P.  X.  P. )
62 0npr 9361 . . . . . . . . 9  |-  -.  (/)  e.  P.
63 ltapr 9414 . . . . . . . . 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 6443 . . . . . . . 8  |-  ( ( w  +P.  ( x  +P.  u ) ) 
<P  ( w  +P.  (
y  +P.  v )
)  ->  ( x  +P.  u )  <P  (
y  +P.  v )
)
65 vex 3111 . . . . . . . . . 10  |-  x  e. 
_V
66 vex 3111 . . . . . . . . . 10  |-  w  e. 
_V
67 addasspr 9391 . . . . . . . . . 10  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
6865, 66, 47, 48, 67caov12 6480 . . . . . . . . 9  |-  ( x  +P.  ( w  +P.  u ) )  =  ( w  +P.  (
x  +P.  u )
)
69 vex 3111 . . . . . . . . . 10  |-  y  e. 
_V
70 vex 3111 . . . . . . . . . 10  |-  v  e. 
_V
7169, 66, 70, 48, 67caov12 6480 . . . . . . . . 9  |-  ( y  +P.  ( w  +P.  v ) )  =  ( w  +P.  (
y  +P.  v )
)
7268, 71breq12i 4451 . . . . . . . 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 9445 . . . . . . . 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 1010 . . 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 7395 . 2  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
( f  <R  g  /\  g  <R  h )  ->  f  <R  h
) )
8033, 79isso2i 4827 1  |-  <R  Or  R.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762   <.cop 4028   class class class wbr 4442    Or wor 4794  (class class class)co 6277   [cec 7301   P.cnp 9228    +P. cpp 9230    <P cltp 9232    ~R cer 9233   R.cnr 9234    <R cltr 9240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-er 7303  df-ec 7305  df-qs 7309  df-ni 9241  df-pli 9242  df-mi 9243  df-lti 9244  df-plpq 9277  df-mpq 9278  df-ltpq 9279  df-enq 9280  df-nq 9281  df-erq 9282  df-plq 9283  df-mq 9284  df-1nq 9285  df-rq 9286  df-ltnq 9287  df-np 9350  df-plp 9352  df-ltp 9354  df-enr 9424  df-nr 9425  df-ltr 9428
This theorem is referenced by:  1ne0sr  9464  addgt0sr  9472  sqgt0sr  9474  supsrlem  9479  axpre-lttri  9533  axpre-lttrn  9534
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